Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 93.9% → 99.0%

Time: 15.7s

Alternatives: 25

Speedup: 1.0×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x\\ \mathbf{if}\;x \leq 10^{-78}:\\ \;\;\;\;t\_0 + \left(\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\right) + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + \left(0.91893853320467 + \frac{0.083333333333333}{x}\right)\right) + \left(z \cdot \left(\frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{x} + z \cdot \frac{y}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x -0.5) (log x))))
   (if (<= x 1e-78)
     (+
      t_0
      (+
       (*
        (/ 1.0 x)
        (+
         0.083333333333333
         (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651))))))
       (- 0.91893853320467 x)))
     (+
      (+ t_0 (+ 0.91893853320467 (/ 0.083333333333333 x)))
      (-
       (*
        z
        (+
         (/ (+ -0.0027777777777778 (* z 0.0007936500793651)) x)
         (* z (/ y x))))
       x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -0.5) * log(x);
	double tmp;
	if (x <= 1e-78) {
		tmp = t_0 + (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + (0.91893853320467 - x));
	} else {
		tmp = (t_0 + (0.91893853320467 + (0.083333333333333 / x))) + ((z * (((-0.0027777777777778 + (z * 0.0007936500793651)) / x) + (z * (y / x)))) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (-0.5d0)) * log(x)
    if (x <= 1d-78) then
        tmp = t_0 + (((1.0d0 / x) * (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0)))))) + (0.91893853320467d0 - x))
    else
        tmp = (t_0 + (0.91893853320467d0 + (0.083333333333333d0 / x))) + ((z * ((((-0.0027777777777778d0) + (z * 0.0007936500793651d0)) / x) + (z * (y / x)))) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + -0.5) * Math.log(x);
	double tmp;
	if (x <= 1e-78) {
		tmp = t_0 + (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + (0.91893853320467 - x));
	} else {
		tmp = (t_0 + (0.91893853320467 + (0.083333333333333 / x))) + ((z * (((-0.0027777777777778 + (z * 0.0007936500793651)) / x) + (z * (y / x)))) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + -0.5) * math.log(x)
	tmp = 0
	if x <= 1e-78:
		tmp = t_0 + (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + (0.91893853320467 - x))
	else:
		tmp = (t_0 + (0.91893853320467 + (0.083333333333333 / x))) + ((z * (((-0.0027777777777778 + (z * 0.0007936500793651)) / x) + (z * (y / x)))) - x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -0.5) * log(x))
	tmp = 0.0
	if (x <= 1e-78)
		tmp = Float64(t_0 + Float64(Float64(Float64(1.0 / x) * Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))))) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(t_0 + Float64(0.91893853320467 + Float64(0.083333333333333 / x))) + Float64(Float64(z * Float64(Float64(Float64(-0.0027777777777778 + Float64(z * 0.0007936500793651)) / x) + Float64(z * Float64(y / x)))) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + -0.5) * log(x);
	tmp = 0.0;
	if (x <= 1e-78)
		tmp = t_0 + (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + (0.91893853320467 - x));
	else
		tmp = (t_0 + (0.91893853320467 + (0.083333333333333 / x))) + ((z * (((-0.0027777777777778 + (z * 0.0007936500793651)) / x) + (z * (y / x)))) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e-78], N[(t$95$0 + N[(N[(N[(1.0 / x), $MachinePrecision] * N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(N[(-0.0027777777777778 + N[(z * 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999999e-79

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, x\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, x\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      10. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \left(x + -0.5\right) \cdot \log x + \left(\color{blue}{\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333\right)} + \left(0.91893853320467 - x\right)\right) \]

   if 9.99999999999999999e-79 < x

   1. Initial program 91.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 91.2%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\log x \cdot \left(x + -0.5\right) + \left(0.91893853320467 + \frac{0.083333333333333}{x}\right)\right) + \left(z \cdot \left(\frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{x} + z \cdot \frac{y}{x}\right) - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-78}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\right) + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x + \left(0.91893853320467 + \frac{0.083333333333333}{x}\right)\right) + \left(z \cdot \left(\frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{x} + z \cdot \frac{y}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x - x\\ \mathbf{if}\;x \leq 5000000000000:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\right) + 0.91893853320467\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ x -0.5) (log x)) x)))
   (if (<= x 5000000000000.0)
     (+
      (+
       (*
        (/ 1.0 x)
        (+
         0.083333333333333
         (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651))))))
       0.91893853320467)
      t_0)
     (+ (* z (* (/ z x) (+ y 0.0007936500793651))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x + -0.5) * log(x)) - x;
	double tmp;
	if (x <= 5000000000000.0) {
		tmp = (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + 0.91893853320467) + t_0;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + (-0.5d0)) * log(x)) - x
    if (x <= 5000000000000.0d0) then
        tmp = (((1.0d0 / x) * (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0)))))) + 0.91893853320467d0) + t_0
    else
        tmp = (z * ((z / x) * (y + 0.0007936500793651d0))) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x + -0.5) * Math.log(x)) - x;
	double tmp;
	if (x <= 5000000000000.0) {
		tmp = (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + 0.91893853320467) + t_0;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x + -0.5) * math.log(x)) - x
	tmp = 0
	if x <= 5000000000000.0:
		tmp = (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + 0.91893853320467) + t_0
	else:
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + -0.5) * log(x)) - x)
	tmp = 0.0
	if (x <= 5000000000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / x) * Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))))) + 0.91893853320467) + t_0);
	else
		tmp = Float64(Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x + -0.5) * log(x)) - x;
	tmp = 0.0;
	if (x <= 5000000000000.0)
		tmp = (((1.0 / x) * (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))))) + 0.91893853320467) + t_0;
	else
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 5000000000000.0], N[(N[(N[(N[(1.0 / x), $MachinePrecision] * N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e12

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 99.7%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   if 5e12 < x

   1. Initial program 88.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 88.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 88.9%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \frac{z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       14. +-lowering-+.f64 99.6%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000000000:\\ \;\;\;\;\left(\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\right) + 0.91893853320467\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x - x\\ \mathbf{if}\;x \leq 8600000000000:\\ \;\;\;\;\left(0.91893853320467 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ x -0.5) (log x)) x)))
   (if (<= x 8600000000000.0)
     (+
      (+
       0.91893853320467
       (/
        (+
         0.083333333333333
         (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
        x))
      t_0)
     (+ (* z (* (/ z x) (+ y 0.0007936500793651))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x + -0.5) * log(x)) - x;
	double tmp;
	if (x <= 8600000000000.0) {
		tmp = (0.91893853320467 + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x)) + t_0;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + (-0.5d0)) * log(x)) - x
    if (x <= 8600000000000.0d0) then
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x)) + t_0
    else
        tmp = (z * ((z / x) * (y + 0.0007936500793651d0))) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x + -0.5) * Math.log(x)) - x;
	double tmp;
	if (x <= 8600000000000.0) {
		tmp = (0.91893853320467 + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x)) + t_0;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x + -0.5) * math.log(x)) - x
	tmp = 0
	if x <= 8600000000000.0:
		tmp = (0.91893853320467 + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x)) + t_0
	else:
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + -0.5) * log(x)) - x)
	tmp = 0.0
	if (x <= 8600000000000.0)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x)) + t_0);
	else
		tmp = Float64(Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x + -0.5) * log(x)) - x;
	tmp = 0.0;
	if (x <= 8600000000000.0)
		tmp = (0.91893853320467 + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x)) + t_0;
	else
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 8600000000000.0], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.6e12

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

   if 8.6e12 < x

   1. Initial program 88.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 88.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 88.9%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \frac{z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       14. +-lowering-+.f64 99.6%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8600000000000:\\ \;\;\;\;\left(0.91893853320467 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x\\ \mathbf{if}\;x \leq 9000000000000:\\ \;\;\;\;t\_0 + \left(\left(0.91893853320467 - x\right) + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(t\_0 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x -0.5) (log x))))
   (if (<= x 9000000000000.0)
     (+
      t_0
      (+
       (- 0.91893853320467 x)
       (/
        (+
         0.083333333333333
         (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
        x)))
     (+ (* z (* (/ z x) (+ y 0.0007936500793651))) (- t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -0.5) * log(x);
	double tmp;
	if (x <= 9000000000000.0) {
		tmp = t_0 + ((0.91893853320467 - x) + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x));
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (t_0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (-0.5d0)) * log(x)
    if (x <= 9000000000000.0d0) then
        tmp = t_0 + ((0.91893853320467d0 - x) + ((0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x))
    else
        tmp = (z * ((z / x) * (y + 0.0007936500793651d0))) + (t_0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + -0.5) * Math.log(x);
	double tmp;
	if (x <= 9000000000000.0) {
		tmp = t_0 + ((0.91893853320467 - x) + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x));
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (t_0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + -0.5) * math.log(x)
	tmp = 0
	if x <= 9000000000000.0:
		tmp = t_0 + ((0.91893853320467 - x) + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x))
	else:
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (t_0 - x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -0.5) * log(x))
	tmp = 0.0
	if (x <= 9000000000000.0)
		tmp = Float64(t_0 + Float64(Float64(0.91893853320467 - x) + Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x)));
	else
		tmp = Float64(Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651))) + Float64(t_0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + -0.5) * log(x);
	tmp = 0.0;
	if (x <= 9000000000000.0)
		tmp = t_0 + ((0.91893853320467 - x) + ((0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x));
	else
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (t_0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9000000000000.0], N[(t$95$0 + N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9e12

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   if 9e12 < x

   1. Initial program 88.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 88.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 88.9%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \frac{z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       14. +-lowering-+.f64 99.6%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9000000000000:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\left(0.91893853320467 - x\right) + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0125:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot \left(0.91893853320467 + -0.5 \cdot \log x\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0125)
   (/
    (+
     0.083333333333333
     (+
      (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651))))
      (* x (+ 0.91893853320467 (* -0.5 (log x))))))
    x)
   (+
    (* z (* (/ z x) (+ y 0.0007936500793651)))
    (- (* (+ x -0.5) (log x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0125) {
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * (0.91893853320467 + (-0.5 * log(x)))))) / x;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * log(x)) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.0125d0) then
        tmp = (0.083333333333333d0 + ((z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0)))) + (x * (0.91893853320467d0 + ((-0.5d0) * log(x)))))) / x
    else
        tmp = (z * ((z / x) * (y + 0.0007936500793651d0))) + (((x + (-0.5d0)) * log(x)) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0125) {
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * (0.91893853320467 + (-0.5 * Math.log(x)))))) / x;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * Math.log(x)) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0125:
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * (0.91893853320467 + (-0.5 * math.log(x)))))) / x
	else:
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * math.log(x)) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0125)
		tmp = Float64(Float64(0.083333333333333 + Float64(Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))) + Float64(x * Float64(0.91893853320467 + Float64(-0.5 * log(x)))))) / x);
	else
		tmp = Float64(Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651))) + Float64(Float64(Float64(x + -0.5) * log(x)) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0125)
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * (0.91893853320467 + (-0.5 * log(x)))))) / x;
	else
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * log(x)) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0125], N[(N[(0.083333333333333 + N[(N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012500000000000001

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), \color{blue}{x}\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + \left(z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right) + x \cdot \left(0.91893853320467 + -0.5 \cdot \log x\right)\right)}{x}} \]

   if 0.012500000000000001 < x

   1. Initial program 89.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 89.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 89.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 89.5%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \frac{z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       14. +-lowering-+.f64 98.8%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   11. Simplified 98.8%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0125:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot \left(0.91893853320467 + -0.5 \cdot \log x\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{1}{x} \cdot \left(0.0007936500793651 + \frac{-0.0027777777777778}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e-16)
   (*
    z
    (*
     z
     (+
      (/ y x)
      (* (/ 1.0 x) (+ 0.0007936500793651 (/ -0.0027777777777778 z))))))
   (if (<= z 4.5e+16)
     (+
      (+ 0.91893853320467 (/ 0.083333333333333 x))
      (- (* (+ x -0.5) (log x)) x))
     (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-16) {
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))));
	} else if (z <= 4.5e+16) {
		tmp = (0.91893853320467 + (0.083333333333333 / x)) + (((x + -0.5) * log(x)) - x);
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8d-16)) then
        tmp = z * (z * ((y / x) + ((1.0d0 / x) * (0.0007936500793651d0 + ((-0.0027777777777778d0) / z)))))
    else if (z <= 4.5d+16) then
        tmp = (0.91893853320467d0 + (0.083333333333333d0 / x)) + (((x + (-0.5d0)) * log(x)) - x)
    else
        tmp = z * ((z * (y + 0.0007936500793651d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-16) {
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))));
	} else if (z <= 4.5e+16) {
		tmp = (0.91893853320467 + (0.083333333333333 / x)) + (((x + -0.5) * Math.log(x)) - x);
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e-16:
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))))
	elif z <= 4.5e+16:
		tmp = (0.91893853320467 + (0.083333333333333 / x)) + (((x + -0.5) * math.log(x)) - x)
	else:
		tmp = z * ((z * (y + 0.0007936500793651)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e-16)
		tmp = Float64(z * Float64(z * Float64(Float64(y / x) + Float64(Float64(1.0 / x) * Float64(0.0007936500793651 + Float64(-0.0027777777777778 / z))))));
	elseif (z <= 4.5e+16)
		tmp = Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) + Float64(Float64(Float64(x + -0.5) * log(x)) - x));
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8e-16)
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))));
	elseif (z <= 4.5e+16)
		tmp = (0.91893853320467 + (0.083333333333333 / x)) + (((x + -0.5) * log(x)) - x);
	else
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e-16], N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(0.0007936500793651 + N[(-0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+16], N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.9999999999999998e-16

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000} \cdot 1}{\color{blue}{x \cdot z}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{\color{blue}{x} \cdot z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right)\right)\right) \]

      8. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{\frac{13888888888889}{5000000000000000}}{x}}{\color{blue}{z}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}{z}\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}}{z}\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{1}{x} \cdot \frac{13888888888889}{5000000000000000}}{z}\right)\right)\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{1}{x} \cdot \color{blue}{\frac{\frac{13888888888889}{5000000000000000}}{z}}\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{1}{x} \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot 1}{z}\right)\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{1}{x} \cdot \left(\frac{13888888888889}{5000000000000000} \cdot \color{blue}{\frac{1}{z}}\right)\right)\right)\right)\right) \]

   7. Simplified 79.8%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{1}{x} \cdot \left(0.0007936500793651 + \frac{-0.0027777777777778}{z}\right)\right)\right)} \]

   if -7.9999999999999998e-16 < z < 4.5e16

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right)}, \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 94.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   9. Simplified 94.7%

      \[\leadsto \left(\color{blue}{\frac{0.083333333333333}{x}} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   if 4.5e16 < z

   1. Initial program 84.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 84.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 90.6%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 90.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 90.7%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{1}{x} \cdot \left(0.0007936500793651 + \frac{-0.0027777777777778}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{1}{x} \cdot \left(0.0007936500793651 + \frac{-0.0027777777777778}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e-16)
   (*
    z
    (*
     z
     (+
      (/ y x)
      (* (/ 1.0 x) (+ 0.0007936500793651 (/ -0.0027777777777778 z))))))
   (if (<= z 5.6e+16)
     (+
      (* (+ x -0.5) (log x))
      (- (+ 0.91893853320467 (/ 0.083333333333333 x)) x))
     (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-16) {
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))));
	} else if (z <= 5.6e+16) {
		tmp = ((x + -0.5) * log(x)) + ((0.91893853320467 + (0.083333333333333 / x)) - x);
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8d-16)) then
        tmp = z * (z * ((y / x) + ((1.0d0 / x) * (0.0007936500793651d0 + ((-0.0027777777777778d0) / z)))))
    else if (z <= 5.6d+16) then
        tmp = ((x + (-0.5d0)) * log(x)) + ((0.91893853320467d0 + (0.083333333333333d0 / x)) - x)
    else
        tmp = z * ((z * (y + 0.0007936500793651d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-16) {
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))));
	} else if (z <= 5.6e+16) {
		tmp = ((x + -0.5) * Math.log(x)) + ((0.91893853320467 + (0.083333333333333 / x)) - x);
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e-16:
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))))
	elif z <= 5.6e+16:
		tmp = ((x + -0.5) * math.log(x)) + ((0.91893853320467 + (0.083333333333333 / x)) - x)
	else:
		tmp = z * ((z * (y + 0.0007936500793651)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e-16)
		tmp = Float64(z * Float64(z * Float64(Float64(y / x) + Float64(Float64(1.0 / x) * Float64(0.0007936500793651 + Float64(-0.0027777777777778 / z))))));
	elseif (z <= 5.6e+16)
		tmp = Float64(Float64(Float64(x + -0.5) * log(x)) + Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) - x));
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8e-16)
		tmp = z * (z * ((y / x) + ((1.0 / x) * (0.0007936500793651 + (-0.0027777777777778 / z)))));
	elseif (z <= 5.6e+16)
		tmp = ((x + -0.5) * log(x)) + ((0.91893853320467 + (0.083333333333333 / x)) - x);
	else
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e-16], N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(0.0007936500793651 + N[(-0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+16], N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.9999999999999998e-16

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000} \cdot 1}{\color{blue}{x \cdot z}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{\color{blue}{x} \cdot z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right)\right)\right) \]

      8. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{y}{x} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{\frac{13888888888889}{5000000000000000}}{x}}{\color{blue}{z}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}{z}\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}}{z}\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{\frac{1}{x} \cdot \frac{13888888888889}{5000000000000000}}{z}\right)\right)\right)\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{1}{x} \cdot \color{blue}{\frac{\frac{13888888888889}{5000000000000000}}{z}}\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{1}{x} \cdot \frac{\frac{13888888888889}{5000000000000000} \cdot 1}{z}\right)\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{1}{x} \cdot \frac{7936500793651}{10000000000000000} - \frac{1}{x} \cdot \left(\frac{13888888888889}{5000000000000000} \cdot \color{blue}{\frac{1}{z}}\right)\right)\right)\right)\right) \]

   7. Simplified 79.8%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{1}{x} \cdot \left(0.0007936500793651 + \frac{-0.0027777777777778}{z}\right)\right)\right)} \]

   if -7.9999999999999998e-16 < z < 5.6e16

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right) - x \]

      2. +-commutative N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)\right) - x \]

      3. associate--l+ N/A

         \[\leadsto \log x \cdot \left(x - \frac{1}{2}\right) + \color{blue}{\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\log x \cdot \left(x - \frac{1}{2}\right)\right), \color{blue}{\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)}\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) \cdot \left(x - \frac{1}{2}\right)\right), \left(\left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      6. log-rec N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) \cdot \left(x - \frac{1}{2}\right)\right), \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right) \cdot \left(x - \frac{1}{2}\right)\right), \left(\left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \left(x - \frac{1}{2}\right)\right), \left(\color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} - x\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \left(x - \frac{1}{2}\right)\right), \left(\left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      10. log-rec N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \left(x - \frac{1}{2}\right)\right), \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\log x, \left(x - \frac{1}{2}\right)\right), \left(\left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      12. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \left(x - \frac{1}{2}\right)\right), \left(\left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) - x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \left(x + \frac{-1}{2}\right)\right), \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) - x\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \left(\left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}}\right) - x\right)\right) \]

      16. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right), \color{blue}{x}\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)\right), x\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right)\right), x\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right)\right), x\right)\right) \]

      20. /-lowering-/.f64 94.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{+.f64}\left(x, \frac{-1}{2}\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right)\right), x\right)\right) \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{\log x \cdot \left(x + -0.5\right) + \left(\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)} \]

   if 5.6e16 < z

   1. Initial program 84.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 84.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 90.6%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 90.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 90.7%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{1}{x} \cdot \left(0.0007936500793651 + \frac{-0.0027777777777778}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0125:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0125)
   (/
    (+
     0.083333333333333
     (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
    x)
   (+
    (* z (* (/ z x) (+ y 0.0007936500793651)))
    (- (* (+ x -0.5) (log x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0125) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * log(x)) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.0125d0) then
        tmp = (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x
    else
        tmp = (z * ((z / x) * (y + 0.0007936500793651d0))) + (((x + (-0.5d0)) * log(x)) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0125) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * Math.log(x)) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0125:
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x
	else:
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * math.log(x)) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0125)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x);
	else
		tmp = Float64(Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651))) + Float64(Float64(Float64(x + -0.5) * log(x)) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0125)
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	else
		tmp = (z * ((z / x) * (y + 0.0007936500793651))) + (((x + -0.5) * log(x)) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0125], N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012500000000000001

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right), \color{blue}{x}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      8. +-lowering-+.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right)}{x}} \]

   if 0.012500000000000001 < x

   1. Initial program 89.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 89.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 89.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 89.5%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y}{x} \cdot z\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \frac{z}{x}\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

       14. +-lowering-+.f64 98.8%

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   11. Simplified 98.8%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0125:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 16.5:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\left(0.91893853320467 - x\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 16.5)
   (/
    (+
     0.083333333333333
     (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
    x)
   (+ (* (+ x -0.5) (log x)) (+ (- 0.91893853320467 x) (* (/ z x) (* z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 16.5) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = ((x + -0.5) * log(x)) + ((0.91893853320467 - x) + ((z / x) * (z * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 16.5d0) then
        tmp = (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x
    else
        tmp = ((x + (-0.5d0)) * log(x)) + ((0.91893853320467d0 - x) + ((z / x) * (z * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 16.5) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = ((x + -0.5) * Math.log(x)) + ((0.91893853320467 - x) + ((z / x) * (z * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 16.5:
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x
	else:
		tmp = ((x + -0.5) * math.log(x)) + ((0.91893853320467 - x) + ((z / x) * (z * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 16.5)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x);
	else
		tmp = Float64(Float64(Float64(x + -0.5) * log(x)) + Float64(Float64(0.91893853320467 - x) + Float64(Float64(z / x) * Float64(z * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 16.5)
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	else
		tmp = ((x + -0.5) * log(x)) + ((0.91893853320467 - x) + ((z / x) * (z * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 16.5], N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 16.5

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right), \color{blue}{x}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      8. +-lowering-+.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right)}{x}} \]

   if 16.5 < x

   1. Initial program 89.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{y \cdot {z}^{2}}{x}\right)}, \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), x\right), \mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, x\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      4. *-lowering-*.f64 82.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

   7. Simplified 82.3%

      \[\leadsto \left(x + -0.5\right) \cdot \log x + \left(\color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} + \left(0.91893853320467 - x\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\left(\frac{\left(y \cdot z\right) \cdot z}{x}\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\left(\left(y \cdot z\right) \cdot \frac{z}{x}\right), \mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(y \cdot z\right), \left(\frac{z}{x}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(z \cdot y\right), \left(\frac{z}{x}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\frac{z}{x}\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

      6. /-lowering-/.f64 89.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{/.f64}\left(z, x\right)\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, x\right)\right)\right) \]

   9. Applied egg-rr 89.6%

      \[\leadsto \left(x + -0.5\right) \cdot \log x + \left(\color{blue}{\left(z \cdot y\right) \cdot \frac{z}{x}} + \left(0.91893853320467 - x\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 16.5:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5\right) \cdot \log x + \left(\left(0.91893853320467 - x\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 920:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot y\right)}{x} + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 920.0)
   (/
    (+
     0.083333333333333
     (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
    x)
   (+ (/ (* z (* z y)) x) (- (* (+ x -0.5) (log x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 920.0) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = ((z * (z * y)) / x) + (((x + -0.5) * log(x)) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 920.0d0) then
        tmp = (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x
    else
        tmp = ((z * (z * y)) / x) + (((x + (-0.5d0)) * log(x)) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 920.0) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = ((z * (z * y)) / x) + (((x + -0.5) * Math.log(x)) - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 920.0:
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x
	else:
		tmp = ((z * (z * y)) / x) + (((x + -0.5) * math.log(x)) - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 920.0)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x);
	else
		tmp = Float64(Float64(Float64(z * Float64(z * y)) / x) + Float64(Float64(Float64(x + -0.5) * log(x)) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 920.0)
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	else
		tmp = ((z * (z * y)) / x) + (((x + -0.5) * log(x)) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 920.0], N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 920

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right), \color{blue}{x}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      8. +-lowering-+.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right)}{x}} \]

   if 920 < x

   1. Initial program 89.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{y \cdot {z}^{2}}{x}\right)}, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), x\right), \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({z}^{2} \cdot y\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(z \cdot z\right) \cdot y\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(z \cdot y\right)\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(y \cdot z\right)\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y \cdot z\right)\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot y\right)\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. *-lowering-*.f64 85.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), x\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   9. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{z \cdot \left(z \cdot y\right)}{x}} - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 920:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot y\right)}{x} + \left(\left(x + -0.5\right) \cdot \log x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + y \cdot \left(z \cdot \left(z + \frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{y}\right)\right)}{x}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+107}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{y}{x} + \frac{z}{\frac{x}{0.0007936500793651}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.75e+32)
   (/
    (+
     0.083333333333333
     (* y (* z (+ z (/ (+ -0.0027777777777778 (* z 0.0007936500793651)) y)))))
    x)
   (if (<= x 7.8e+107)
     (* z (+ (* z (/ y x)) (/ z (/ x 0.0007936500793651))))
     (* x (+ (log x) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.75e+32) {
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x;
	} else if (x <= 7.8e+107) {
		tmp = z * ((z * (y / x)) + (z / (x / 0.0007936500793651)));
	} else {
		tmp = x * (log(x) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.75d+32) then
        tmp = (0.083333333333333d0 + (y * (z * (z + (((-0.0027777777777778d0) + (z * 0.0007936500793651d0)) / y))))) / x
    else if (x <= 7.8d+107) then
        tmp = z * ((z * (y / x)) + (z / (x / 0.0007936500793651d0)))
    else
        tmp = x * (log(x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.75e+32) {
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x;
	} else if (x <= 7.8e+107) {
		tmp = z * ((z * (y / x)) + (z / (x / 0.0007936500793651)));
	} else {
		tmp = x * (Math.log(x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.75e+32:
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x
	elif x <= 7.8e+107:
		tmp = z * ((z * (y / x)) + (z / (x / 0.0007936500793651)))
	else:
		tmp = x * (math.log(x) + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.75e+32)
		tmp = Float64(Float64(0.083333333333333 + Float64(y * Float64(z * Float64(z + Float64(Float64(-0.0027777777777778 + Float64(z * 0.0007936500793651)) / y))))) / x);
	elseif (x <= 7.8e+107)
		tmp = Float64(z * Float64(Float64(z * Float64(y / x)) + Float64(z / Float64(x / 0.0007936500793651))));
	else
		tmp = Float64(x * Float64(log(x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.75e+32)
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x;
	elseif (x <= 7.8e+107)
		tmp = z * ((z * (y / x)) + (z / (x / 0.0007936500793651)));
	else
		tmp = x * (log(x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.75e+32], N[(N[(0.083333333333333 + N[(y * N[(z * N[(z + N[(N[(-0.0027777777777778 + N[(z * 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.8e+107], N[(z * N[(N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(z / N[(x / 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.75e32

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x}{y} + -1 \cdot \frac{{z}^{2}}{x}\right)\right)} \]

   7. Simplified 69.8%

      \[\leadsto \color{blue}{y \cdot \left(\frac{z \cdot z}{x} + \frac{0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x + -0.5\right) + \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right) - x\right)\right)}{y}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)\right), \color{blue}{x}\right) \]

   10. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{0.083333333333333 + y \cdot \left(z \cdot \left(z + \frac{z \cdot 0.0007936500793651 + -0.0027777777777778}{y}\right)\right)}{x}} \]

   if 1.75e32 < x < 7.7999999999999997e107

   1. Initial program 85.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 85.8%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 64.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 64.3%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{z \cdot \frac{y}{x}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \color{blue}{z} \cdot \frac{y}{x}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z\right), \color{blue}{\left(z \cdot \frac{y}{x}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\color{blue}{z} \cdot \frac{y}{x}\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \frac{1}{\frac{x}{\frac{7936500793651}{10000000000000000}}}\right), \left(z \cdot \frac{y}{x}\right)\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{z}{\frac{x}{\frac{7936500793651}{10000000000000000}}}\right), \left(\color{blue}{z} \cdot \frac{y}{x}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{x}{\frac{7936500793651}{10000000000000000}}\right)\right), \left(\color{blue}{z} \cdot \frac{y}{x}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \frac{7936500793651}{10000000000000000}\right)\right), \left(z \cdot \frac{y}{x}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \frac{7936500793651}{10000000000000000}\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      10. /-lowering-/.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \frac{7936500793651}{10000000000000000}\right)\right), \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   9. Applied egg-rr 64.4%

      \[\leadsto z \cdot \color{blue}{\left(\frac{z}{\frac{x}{0.0007936500793651}} + z \cdot \frac{y}{x}\right)} \]

   if 7.7999999999999997e107 < x

   1. Initial program 89.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)\right)\right) \]

      7. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \log x\right)\right) \]

      9. log-lowering-log.f64 88.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{log.f64}\left(x\right)\right)\right) \]

   7. Simplified 88.1%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + y \cdot \left(z \cdot \left(z + \frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{y}\right)\right)}{x}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+107}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{y}{x} + \frac{z}{\frac{x}{0.0007936500793651}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + y \cdot \left(z \cdot \left(z + \frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{y}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.2e+32)
   (/
    (+
     0.083333333333333
     (* y (* z (+ z (/ (+ -0.0027777777777778 (* z 0.0007936500793651)) y)))))
    x)
   (* z (/ (* z (+ y 0.0007936500793651)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.2e+32) {
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5.2d+32) then
        tmp = (0.083333333333333d0 + (y * (z * (z + (((-0.0027777777777778d0) + (z * 0.0007936500793651d0)) / y))))) / x
    else
        tmp = z * ((z * (y + 0.0007936500793651d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.2e+32) {
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5.2e+32:
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x
	else:
		tmp = z * ((z * (y + 0.0007936500793651)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.2e+32)
		tmp = Float64(Float64(0.083333333333333 + Float64(y * Float64(z * Float64(z + Float64(Float64(-0.0027777777777778 + Float64(z * 0.0007936500793651)) / y))))) / x);
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5.2e+32)
		tmp = (0.083333333333333 + (y * (z * (z + ((-0.0027777777777778 + (z * 0.0007936500793651)) / y))))) / x;
	else
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.2e+32], N[(N[(0.083333333333333 + N[(y * N[(z * N[(z + N[(N[(-0.0027777777777778 + N[(z * 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2000000000000004e32

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x}{y} + -1 \cdot \frac{{z}^{2}}{x}\right)\right)} \]

   7. Simplified 69.8%

      \[\leadsto \color{blue}{y \cdot \left(\frac{z \cdot z}{x} + \frac{0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(\left(\log x \cdot \left(x + -0.5\right) + \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right) - x\right)\right)}{y}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{y} + \left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{y} + {z}^{2}\right)\right)\right), \color{blue}{x}\right) \]

   10. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{0.083333333333333 + y \cdot \left(z \cdot \left(z + \frac{z \cdot 0.0007936500793651 + -0.0027777777777778}{y}\right)\right)}{x}} \]

   if 5.2000000000000004e32 < x

   1. Initial program 88.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 29.4%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 29.5%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + y \cdot \left(z \cdot \left(z + \frac{-0.0027777777777778 + z \cdot 0.0007936500793651}{y}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -110:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -110.0)
   (* z (* (/ z x) (+ y 0.0007936500793651)))
   (if (<= z 2.7e-5)
     (/ (+ 0.083333333333333 (* z (* z y))) x)
     (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -110.0) {
		tmp = z * ((z / x) * (y + 0.0007936500793651));
	} else if (z <= 2.7e-5) {
		tmp = (0.083333333333333 + (z * (z * y))) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-110.0d0)) then
        tmp = z * ((z / x) * (y + 0.0007936500793651d0))
    else if (z <= 2.7d-5) then
        tmp = (0.083333333333333d0 + (z * (z * y))) / x
    else
        tmp = z * ((z * (y + 0.0007936500793651d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -110.0) {
		tmp = z * ((z / x) * (y + 0.0007936500793651));
	} else if (z <= 2.7e-5) {
		tmp = (0.083333333333333 + (z * (z * y))) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -110.0:
		tmp = z * ((z / x) * (y + 0.0007936500793651))
	elif z <= 2.7e-5:
		tmp = (0.083333333333333 + (z * (z * y))) / x
	else:
		tmp = z * ((z * (y + 0.0007936500793651)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -110.0)
		tmp = Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651)));
	elseif (z <= 2.7e-5)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(z * y))) / x);
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -110.0)
		tmp = z * ((z / x) * (y + 0.0007936500793651));
	elseif (z <= 2.7e-5)
		tmp = (0.083333333333333 + (z * (z * y))) / x;
	else
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -110.0], N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e-5], N[(N[(0.083333333333333 + N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -110

   1. Initial program 92.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 92.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 92.7%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

       2. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \color{blue}{\frac{y}{x} \cdot z}\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{\color{blue}{y}}{x} \cdot z\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \color{blue}{\frac{y}{x}} \cdot z\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y}{x}} \cdot z\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{\color{blue}{x}}\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} + y\right)\right)\right) \]

       14. +-lowering-+.f64 80.2%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{y}\right)\right)\right) \]

   11. Simplified 80.2%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} \]

   if -110 < z < 2.6999999999999999e-5

   1. Initial program 99.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   7. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + \left(z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right) + x \cdot \left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right)\right)}{x}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{\left(y \cdot {z}^{2}\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left({z}^{2} \cdot y\right)\right), x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(\left(z \cdot z\right) \cdot y\right)\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot y\right)\right)\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(y \cdot z\right)\right)\right), x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(y \cdot z\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot y\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 55.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right)\right), x\right) \]

   10. Simplified 55.5%

       \[\leadsto \frac{0.083333333333333 + \color{blue}{z \cdot \left(z \cdot y\right)}}{x} \]

   if 2.6999999999999999e-5 < z

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 82.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 82.8%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 82.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 82.8%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -110:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-19}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e-19)
   (* z (* (/ z x) (+ y 0.0007936500793651)))
   (if (<= z 3.1e-13)
     (/ (+ 0.083333333333333 (* z -0.0027777777777778)) x)
     (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e-19) {
		tmp = z * ((z / x) * (y + 0.0007936500793651));
	} else if (z <= 3.1e-13) {
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.9d-19)) then
        tmp = z * ((z / x) * (y + 0.0007936500793651d0))
    else if (z <= 3.1d-13) then
        tmp = (0.083333333333333d0 + (z * (-0.0027777777777778d0))) / x
    else
        tmp = z * ((z * (y + 0.0007936500793651d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e-19) {
		tmp = z * ((z / x) * (y + 0.0007936500793651));
	} else if (z <= 3.1e-13) {
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	} else {
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e-19:
		tmp = z * ((z / x) * (y + 0.0007936500793651))
	elif z <= 3.1e-13:
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x
	else:
		tmp = z * ((z * (y + 0.0007936500793651)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e-19)
		tmp = Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651)));
	elseif (z <= 3.1e-13)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * -0.0027777777777778)) / x);
	else
		tmp = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e-19)
		tmp = z * ((z / x) * (y + 0.0007936500793651));
	elseif (z <= 3.1e-13)
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	else
		tmp = z * ((z * (y + 0.0007936500793651)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e-19], N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e-13], N[(N[(0.083333333333333 + N[(z * -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e-19

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 93.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 93.0%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

       2. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \color{blue}{\frac{y}{x} \cdot z}\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{\color{blue}{y}}{x} \cdot z\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \color{blue}{\frac{y}{x}} \cdot z\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y}{x}} \cdot z\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{\color{blue}{x}}\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} + y\right)\right)\right) \]

       14. +-lowering-+.f64 79.7%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{y}\right)\right)\right) \]

   11. Simplified 79.7%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} \]

   if -1.9e-19 < z < 3.0999999999999999e-13

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)}, \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{\frac{-13888888888889}{5000000000000000} \cdot z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. *-lowering-*.f64 95.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   9. Simplified 95.9%

      \[\leadsto \left(\color{blue}{\left(\frac{0.083333333333333}{x} + \frac{-0.0027777777777778 \cdot z}{x}\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{x}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z\right), \color{blue}{x}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(\frac{-13888888888889}{5000000000000000} \cdot z\right)\right), x\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \frac{-13888888888889}{5000000000000000}\right)\right), x\right) \]

       4. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \frac{-13888888888889}{5000000000000000}\right)\right), x\right) \]

   12. Simplified 51.6%

       \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}} \]

   if 3.0999999999999999e-13 < z

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 82.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 82.8%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 82.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 82.8%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-19}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* (/ z x) (+ y 0.0007936500793651)))))
   (if (<= z -3.5e-17)
     t_0
     (if (<= z 2.8e-10)
       (/ (+ 0.083333333333333 (* z -0.0027777777777778)) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z / x) * (y + 0.0007936500793651));
	double tmp;
	if (z <= -3.5e-17) {
		tmp = t_0;
	} else if (z <= 2.8e-10) {
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((z / x) * (y + 0.0007936500793651d0))
    if (z <= (-3.5d-17)) then
        tmp = t_0
    else if (z <= 2.8d-10) then
        tmp = (0.083333333333333d0 + (z * (-0.0027777777777778d0))) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z / x) * (y + 0.0007936500793651));
	double tmp;
	if (z <= -3.5e-17) {
		tmp = t_0;
	} else if (z <= 2.8e-10) {
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z / x) * (y + 0.0007936500793651))
	tmp = 0
	if z <= -3.5e-17:
		tmp = t_0
	elif z <= 2.8e-10:
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z / x) * Float64(y + 0.0007936500793651)))
	tmp = 0.0
	if (z <= -3.5e-17)
		tmp = t_0;
	elseif (z <= 2.8e-10)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * -0.0027777777777778)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z / x) * (y + 0.0007936500793651));
	tmp = 0.0;
	if (z <= -3.5e-17)
		tmp = t_0;
	elseif (z <= 2.8e-10)
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e-17], t$95$0, If[LessEqual[z, 2.8e-10], N[(N[(0.083333333333333 + N[(z * -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000002e-17 or 2.80000000000000015e-10 < z

   1. Initial program 90.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{x}{z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x}\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      12. +-lowering-+.f64 90.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. Applied egg-rr 90.1%

      \[\leadsto \left(\color{blue}{\frac{1}{x} \cdot \left(0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right)\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

       2. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \color{blue}{\frac{y}{x} \cdot z}\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \cdot z + \frac{\color{blue}{y}}{x} \cdot z\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x} \cdot z + \frac{y}{x} \cdot z\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x} + \color{blue}{\frac{y}{x}} \cdot z\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y}{x}} \cdot z\right)\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \frac{y \cdot z}{\color{blue}{x}}\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + y \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

       11. distribute-rgt-out N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z}{x} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} + y\right)\right)\right) \]

       14. +-lowering-+.f64 81.0%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{y}\right)\right)\right) \]

   11. Simplified 81.0%

       \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right)} \]

   if -3.5000000000000002e-17 < z < 2.80000000000000015e-10

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)}, \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{\frac{-13888888888889}{5000000000000000} \cdot z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. *-lowering-*.f64 95.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   9. Simplified 95.9%

      \[\leadsto \left(\color{blue}{\left(\frac{0.083333333333333}{x} + \frac{-0.0027777777777778 \cdot z}{x}\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{x}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z\right), \color{blue}{x}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(\frac{-13888888888889}{5000000000000000} \cdot z\right)\right), x\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \frac{-13888888888889}{5000000000000000}\right)\right), x\right) \]

       4. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \frac{-13888888888889}{5000000000000000}\right)\right), x\right) \]

   12. Simplified 51.6%

       \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-17}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;x \leq 1.16 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + t\_0\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651))))
   (if (<= x 1.16e+33)
     (/ (+ 0.083333333333333 (* z (+ -0.0027777777777778 t_0))) x)
     (* z (/ t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double tmp;
	if (x <= 1.16e+33) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + t_0))) / x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y + 0.0007936500793651d0)
    if (x <= 1.16d+33) then
        tmp = (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + t_0))) / x
    else
        tmp = z * (t_0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double tmp;
	if (x <= 1.16e+33) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + t_0))) / x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y + 0.0007936500793651)
	tmp = 0
	if x <= 1.16e+33:
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + t_0))) / x
	else:
		tmp = z * (t_0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (x <= 1.16e+33)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + t_0))) / x);
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y + 0.0007936500793651);
	tmp = 0.0;
	if (x <= 1.16e+33)
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + t_0))) / x;
	else
		tmp = z * (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.16e+33], N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16000000000000001e33

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right), \color{blue}{x}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      8. +-lowering-+.f64 95.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

   7. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right)}{x}} \]

   if 1.16000000000000001e33 < x

   1. Initial program 88.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 29.4%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 29.5%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.85 \cdot 10^{-16}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\frac{x}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.85e-16)
   (* (/ z x) (* z y))
   (if (<= z 3.2e-8)
     (/ (+ 0.083333333333333 (* z -0.0027777777777778)) x)
     (* y (/ z (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.85e-16) {
		tmp = (z / x) * (z * y);
	} else if (z <= 3.2e-8) {
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	} else {
		tmp = y * (z / (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.85d-16)) then
        tmp = (z / x) * (z * y)
    else if (z <= 3.2d-8) then
        tmp = (0.083333333333333d0 + (z * (-0.0027777777777778d0))) / x
    else
        tmp = y * (z / (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.85e-16) {
		tmp = (z / x) * (z * y);
	} else if (z <= 3.2e-8) {
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	} else {
		tmp = y * (z / (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.85e-16:
		tmp = (z / x) * (z * y)
	elif z <= 3.2e-8:
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x
	else:
		tmp = y * (z / (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.85e-16)
		tmp = Float64(Float64(z / x) * Float64(z * y));
	elseif (z <= 3.2e-8)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * -0.0027777777777778)) / x);
	else
		tmp = Float64(y * Float64(z / Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.85e-16)
		tmp = (z / x) * (z * y);
	elseif (z <= 3.2e-8)
		tmp = (0.083333333333333 + (z * -0.0027777777777778)) / x;
	else
		tmp = y * (z / (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.85e-16], N[(N[(z / x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e-8], N[(N[(0.083333333333333 + N[(z * -0.0027777777777778), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.84999999999999994e-16

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(y \cdot z\right) \cdot z}{x} \]

      2. associate-/l* N/A

         \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. /-lowering-/.f64 57.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{\left(z \cdot y\right) \cdot \frac{z}{x}} \]

   if -3.84999999999999994e-16 < z < 3.2000000000000002e-8

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. associate-+r- N/A

         \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

      3. associate-+l- N/A

         \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)}, \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{\frac{-13888888888889}{5000000000000000} \cdot z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

      8. *-lowering-*.f64 95.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   9. Simplified 95.9%

      \[\leadsto \left(\color{blue}{\left(\frac{0.083333333333333}{x} + \frac{-0.0027777777777778 \cdot z}{x}\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{x}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z\right), \color{blue}{x}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(\frac{-13888888888889}{5000000000000000} \cdot z\right)\right), x\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \frac{-13888888888889}{5000000000000000}\right)\right), x\right) \]

       4. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \frac{-13888888888889}{5000000000000000}\right)\right), x\right) \]

   12. Simplified 51.6%

       \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}} \]

   if 3.2000000000000002e-8 < z

   1. Initial program 86.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{y} \]

      3. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot y \]

      4. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{z}{x} \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{x} \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{y}\right)\right) \]

      7. /-lowering-/.f64 57.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), y\right)\right) \]

   9. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot \color{blue}{y} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \frac{z}{x}\right), \color{blue}{y}\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \frac{1}{\frac{x}{z}}\right), y\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{z}{\frac{x}{z}}\right), y\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{x}{z}\right)\right), y\right) \]

       6. /-lowering-/.f64 60.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, z\right)\right), y\right) \]

   11. Applied egg-rr 60.8%

       \[\leadsto \color{blue}{\frac{z}{\frac{x}{z}} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.85 \cdot 10^{-16}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot -0.0027777777777778}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{\frac{x}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+16)
   (/ y (/ x (* z z)))
   (if (<= y 0.0008)
     (* z (* z (/ 0.0007936500793651 x)))
     (* (/ z x) (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+16) {
		tmp = y / (x / (z * z));
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = (z / x) * (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.4d+16)) then
        tmp = y / (x / (z * z))
    else if (y <= 0.0008d0) then
        tmp = z * (z * (0.0007936500793651d0 / x))
    else
        tmp = (z / x) * (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+16) {
		tmp = y / (x / (z * z));
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = (z / x) * (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e+16:
		tmp = y / (x / (z * z))
	elif y <= 0.0008:
		tmp = z * (z * (0.0007936500793651 / x))
	else:
		tmp = (z / x) * (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+16)
		tmp = Float64(y / Float64(x / Float64(z * z)));
	elseif (y <= 0.0008)
		tmp = Float64(z * Float64(z * Float64(0.0007936500793651 / x)));
	else
		tmp = Float64(Float64(z / x) * Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e+16)
		tmp = y / (x / (z * z));
	elseif (y <= 0.0008)
		tmp = z * (z * (0.0007936500793651 / x));
	else
		tmp = (z / x) * (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+16], N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0008], N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e16

   1. Initial program 98.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 50.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      2. clear-num N/A

         \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{x}{z \cdot z}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{x}{z \cdot z}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{x}{z \cdot z}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. *-lowering-*.f64 52.4%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{z \cdot z}}} \]

   if -2.4e16 < y < 8.00000000000000038e-4

   1. Initial program 93.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{\color{blue}{x}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 32.5%

       \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{0.0007936500793651}{x}\right)} \]

   if 8.00000000000000038e-4 < y

   1. Initial program 95.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(y \cdot z\right) \cdot z}{x} \]

      2. associate-/l* N/A

         \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. /-lowering-/.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\left(z \cdot y\right) \cdot \frac{z}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \frac{z}{\frac{x}{z}}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+16)
   (* y (/ z (/ x z)))
   (if (<= y 0.0008)
     (* z (* z (/ 0.0007936500793651 x)))
     (* (/ z x) (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+16) {
		tmp = y * (z / (x / z));
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = (z / x) * (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.4d+16)) then
        tmp = y * (z / (x / z))
    else if (y <= 0.0008d0) then
        tmp = z * (z * (0.0007936500793651d0 / x))
    else
        tmp = (z / x) * (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+16) {
		tmp = y * (z / (x / z));
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = (z / x) * (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e+16:
		tmp = y * (z / (x / z))
	elif y <= 0.0008:
		tmp = z * (z * (0.0007936500793651 / x))
	else:
		tmp = (z / x) * (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+16)
		tmp = Float64(y * Float64(z / Float64(x / z)));
	elseif (y <= 0.0008)
		tmp = Float64(z * Float64(z * Float64(0.0007936500793651 / x)));
	else
		tmp = Float64(Float64(z / x) * Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e+16)
		tmp = y * (z / (x / z));
	elseif (y <= 0.0008)
		tmp = z * (z * (0.0007936500793651 / x));
	else
		tmp = (z / x) * (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+16], N[(y * N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0008], N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e16

   1. Initial program 98.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 50.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{y} \]

      3. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot y \]

      4. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{z}{x} \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{x} \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{y}\right)\right) \]

      7. /-lowering-/.f64 52.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), y\right)\right) \]

   9. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot \color{blue}{y} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \frac{z}{x}\right), \color{blue}{y}\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \frac{1}{\frac{x}{z}}\right), y\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{z}{\frac{x}{z}}\right), y\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{x}{z}\right)\right), y\right) \]

       6. /-lowering-/.f64 52.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, z\right)\right), y\right) \]

   11. Applied egg-rr 52.4%

       \[\leadsto \color{blue}{\frac{z}{\frac{x}{z}} \cdot y} \]

   if -2.4e16 < y < 8.00000000000000038e-4

   1. Initial program 93.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{\color{blue}{x}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 32.5%

       \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{0.0007936500793651}{x}\right)} \]

   if 8.00000000000000038e-4 < y

   1. Initial program 95.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(y \cdot z\right) \cdot z}{x} \]

      2. associate-/l* N/A

         \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. /-lowering-/.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\left(z \cdot y\right) \cdot \frac{z}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \frac{z}{\frac{x}{z}}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z x) (* z y))))
   (if (<= y -2.4e+16)
     t_0
     (if (<= y 0.0008) (* z (* z (/ 0.0007936500793651 x))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z / x) * (z * y);
	double tmp;
	if (y <= -2.4e+16) {
		tmp = t_0;
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z / x) * (z * y)
    if (y <= (-2.4d+16)) then
        tmp = t_0
    else if (y <= 0.0008d0) then
        tmp = z * (z * (0.0007936500793651d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z / x) * (z * y);
	double tmp;
	if (y <= -2.4e+16) {
		tmp = t_0;
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / x) * (z * y)
	tmp = 0
	if y <= -2.4e+16:
		tmp = t_0
	elif y <= 0.0008:
		tmp = z * (z * (0.0007936500793651 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / x) * Float64(z * y))
	tmp = 0.0
	if (y <= -2.4e+16)
		tmp = t_0;
	elseif (y <= 0.0008)
		tmp = Float64(z * Float64(z * Float64(0.0007936500793651 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / x) * (z * y);
	tmp = 0.0;
	if (y <= -2.4e+16)
		tmp = t_0;
	elseif (y <= 0.0008)
		tmp = z * (z * (0.0007936500793651 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+16], t$95$0, If[LessEqual[y, 0.0008], N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e16 or 8.00000000000000038e-4 < y

   1. Initial program 96.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 52.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(y \cdot z\right) \cdot z}{x} \]

      2. associate-/l* N/A

         \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{\frac{z}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. /-lowering-/.f64 54.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 54.2%

      \[\leadsto \color{blue}{\left(z \cdot y\right) \cdot \frac{z}{x}} \]

   if -2.4e16 < y < 8.00000000000000038e-4

   1. Initial program 93.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{\color{blue}{x}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 32.5%

       \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{0.0007936500793651}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y (/ z x)))))
   (if (<= y -2.4e+16)
     t_0
     (if (<= y 0.0008) (* z (* z (/ 0.0007936500793651 x))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * (z / x));
	double tmp;
	if (y <= -2.4e+16) {
		tmp = t_0;
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y * (z / x))
    if (y <= (-2.4d+16)) then
        tmp = t_0
    else if (y <= 0.0008d0) then
        tmp = z * (z * (0.0007936500793651d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y * (z / x));
	double tmp;
	if (y <= -2.4e+16) {
		tmp = t_0;
	} else if (y <= 0.0008) {
		tmp = z * (z * (0.0007936500793651 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * (z / x))
	tmp = 0
	if y <= -2.4e+16:
		tmp = t_0
	elif y <= 0.0008:
		tmp = z * (z * (0.0007936500793651 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * Float64(z / x)))
	tmp = 0.0
	if (y <= -2.4e+16)
		tmp = t_0;
	elseif (y <= 0.0008)
		tmp = Float64(z * Float64(z * Float64(0.0007936500793651 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * (z / x));
	tmp = 0.0;
	if (y <= -2.4e+16)
		tmp = t_0;
	elseif (y <= 0.0008)
		tmp = z * (z * (0.0007936500793651 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+16], t$95$0, If[LessEqual[y, 0.0008], N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e16 or 8.00000000000000038e-4 < y

   1. Initial program 96.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{x}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left({z}^{2}\right)\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

      4. *-lowering-*.f64 52.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{z \cdot z}{x} \cdot \color{blue}{y} \]

      3. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot y \]

      4. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{z}{x} \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{x} \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(\frac{z}{x}\right), \color{blue}{y}\right)\right) \]

      7. /-lowering-/.f64 53.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, x\right), y\right)\right) \]

   9. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{z \cdot \left(\frac{z}{x} \cdot y\right)} \]

   if -2.4e16 < y < 8.00000000000000038e-4

   1. Initial program 93.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{\color{blue}{x}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right)\right)\right) \]

      9. /-lowering-/.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 32.5%

       \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{0.0007936500793651}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651))))
   (if (<= x 9.5e+32) (/ (+ 0.083333333333333 (* z t_0)) x) (* z (/ t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double tmp;
	if (x <= 9.5e+32) {
		tmp = (0.083333333333333 + (z * t_0)) / x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y + 0.0007936500793651d0)
    if (x <= 9.5d+32) then
        tmp = (0.083333333333333d0 + (z * t_0)) / x
    else
        tmp = z * (t_0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double tmp;
	if (x <= 9.5e+32) {
		tmp = (0.083333333333333 + (z * t_0)) / x;
	} else {
		tmp = z * (t_0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y + 0.0007936500793651)
	tmp = 0
	if x <= 9.5e+32:
		tmp = (0.083333333333333 + (z * t_0)) / x
	else:
		tmp = z * (t_0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (x <= 9.5e+32)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * t_0)) / x);
	else
		tmp = Float64(z * Float64(t_0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y + 0.0007936500793651);
	tmp = 0.0;
	if (x <= 9.5e+32)
		tmp = (0.083333333333333 + (z * t_0)) / x;
	else
		tmp = z * (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.5e+32], N[(N[(0.083333333333333 + N[(z * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.50000000000000006e32

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + \left(z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right) + x \cdot \left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right)\right)}{x}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{\left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(\left(z \cdot z\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right), x\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right)\right)\right), x\right) \]

      5. +-lowering-+.f64 94.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right)\right), x\right) \]

   10. Simplified 94.7%

       \[\leadsto \frac{0.083333333333333 + \color{blue}{z \cdot \left(z \cdot \left(0.0007936500793651 + y\right)\right)}}{x} \]

   if 9.50000000000000006e32 < x

   1. Initial program 88.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. associate-+l- N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 29.4%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} + y}}{x} \]

      3. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-/l* N/A

         \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \color{blue}{x}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), x\right)\right) \]

      8. +-lowering-+.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   10. Simplified 29.5%

       \[\leadsto \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 26.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z (* z (/ 0.0007936500793651 x))))

C

double code(double x, double y, double z) {
	return z * (z * (0.0007936500793651 / x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * (z * (0.0007936500793651d0 / x))
end function

Java

public static double code(double x, double y, double z) {
	return z * (z * (0.0007936500793651 / x));
}

Python

def code(x, y, z):
	return z * (z * (0.0007936500793651 / x))

Julia

function code(x, y, z)
	return Float64(z * Float64(z * Float64(0.0007936500793651 / x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * (z * (0.0007936500793651 / x));
end

Wolfram

code[x_, y_, z_] := N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   2. associate-+l- N/A

      \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

   10. associate--r+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

   2. associate-*l* N/A

      \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right), \left(\frac{y}{x}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 40.9%

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

7. Simplified 40.9%

   \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{\color{blue}{x}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{x}\right)\right)\right) \]

   9. /-lowering-/.f64 26.9%

      \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{x}\right)\right)\right) \]

10. Simplified 26.9%

    \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{0.0007936500793651}{x}\right)} \]
11. Add Preprocessing

Alternative 24: 8.4% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{-0.0027777777777778}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z (/ -0.0027777777777778 x)))

C

double code(double x, double y, double z) {
	return z * (-0.0027777777777778 / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * ((-0.0027777777777778d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return z * (-0.0027777777777778 / x);
}

Python

def code(x, y, z):
	return z * (-0.0027777777777778 / x)

Julia

function code(x, y, z)
	return Float64(z * Float64(-0.0027777777777778 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * (-0.0027777777777778 / x);
end

Wolfram

code[x_, y_, z_] := N[(z * N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   2. associate-+l- N/A

      \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

   10. associate--r+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

   2. associate-+r- N/A

      \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

   3. associate-+l- N/A

      \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

6. Applied egg-rr 95.0%

   \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)}, \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{\frac{-13888888888889}{5000000000000000} \cdot z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. *-lowering-*.f64 67.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

9. Simplified 67.3%

   \[\leadsto \left(\color{blue}{\left(\frac{0.083333333333333}{x} + \frac{-0.0027777777777778 \cdot z}{x}\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

10. Taylor expanded in z around inf

    \[\leadsto \color{blue}{\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), \color{blue}{x}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-13888888888889}{5000000000000000}\right), x\right) \]

    4. *-lowering-*.f64 10.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-13888888888889}{5000000000000000}\right), x\right) \]

12. Simplified 10.1%

    \[\leadsto \color{blue}{\frac{z \cdot -0.0027777777777778}{x}} \]
13. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto z \cdot \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{-13888888888889}{5000000000000000}}{x} \cdot \color{blue}{z} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{z}\right) \]

    4. /-lowering-/.f64 10.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-13888888888889}{5000000000000000}, x\right), z\right) \]

14. Applied egg-rr 10.1%

    \[\leadsto \color{blue}{\frac{-0.0027777777777778}{x} \cdot z} \]

15. Final simplification 10.1%

    \[\leadsto z \cdot \frac{-0.0027777777777778}{x} \]
16. Add Preprocessing

Alternative 25: 8.5% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ -0.0027777777777778 \cdot \frac{z}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -0.0027777777777778 (/ z x)))

C

double code(double x, double y, double z) {
	return -0.0027777777777778 * (z / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-0.0027777777777778d0) * (z / x)
end function

Java

public static double code(double x, double y, double z) {
	return -0.0027777777777778 * (z / x);
}

Python

def code(x, y, z):
	return -0.0027777777777778 * (z / x)

Julia

function code(x, y, z)
	return Float64(-0.0027777777777778 * Float64(z / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.0027777777777778 * (z / x);
end

Wolfram

code[x_, y_, z_] := N[(-0.0027777777777778 * N[(z / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   2. associate-+l- N/A

      \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x - \color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), \left(\mathsf{neg}\left(\left(\color{blue}{x} - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(\mathsf{neg}\left(\left(x - \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(x - \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right)\right)\right)\right) \]

   10. associate--r+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) - \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(x - \frac{91893853320467}{100000000000000}\right) + \left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) + \left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{91893853320467}{100000000000000}\right)\right)\right)}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\mathsf{neg}\left(\color{blue}{\left(x - \frac{91893853320467}{100000000000000}\right)}\right)\right)\right)\right) \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot \log x + \left(\frac{z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 - x\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) + \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x} \]

   2. associate-+r- N/A

      \[\leadsto \left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) + \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x \]

   3. associate-+l- N/A

      \[\leadsto \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right), \color{blue}{\left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x}\right), \frac{91893853320467}{100000000000000}\right), \left(\color{blue}{x} - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \left(x - \left(x + \frac{-1}{2}\right) \cdot \log x\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), x\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right)}\right)\right) \]

6. Applied egg-rr 95.0%

   \[\leadsto \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) + 0.083333333333333}{x} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)}, \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \left(\frac{\frac{-13888888888889}{5000000000000000} \cdot z}{x}\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   8. *-lowering-*.f64 67.3%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, z\right), x\right)\right), \frac{91893853320467}{100000000000000}\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

9. Simplified 67.3%

   \[\leadsto \left(\color{blue}{\left(\frac{0.083333333333333}{x} + \frac{-0.0027777777777778 \cdot z}{x}\right)} + 0.91893853320467\right) - \left(x - \left(x + -0.5\right) \cdot \log x\right) \]

10. Taylor expanded in z around inf

    \[\leadsto \color{blue}{\frac{-13888888888889}{5000000000000000} \cdot \frac{z}{x}} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z}{\color{blue}{x}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-13888888888889}{5000000000000000} \cdot z\right), \color{blue}{x}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \frac{-13888888888889}{5000000000000000}\right), x\right) \]

    4. *-lowering-*.f64 10.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \frac{-13888888888889}{5000000000000000}\right), x\right) \]

12. Simplified 10.1%

    \[\leadsto \color{blue}{\frac{z \cdot -0.0027777777777778}{x}} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{-13888888888889}{5000000000000000} \cdot z}{x} \]

    2. associate-/l* N/A

       \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

    4. /-lowering-/.f64 10.1%

       \[\leadsto \mathsf{*.f64}\left(\frac{-13888888888889}{5000000000000000}, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

14. Applied egg-rr 10.1%

    \[\leadsto \color{blue}{-0.0027777777777778 \cdot \frac{z}{x}} \]
15. Add Preprocessing

Developer Target 1: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x))
  (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) + (0.91893853320467d0 - x)) + (0.083333333333333d0 / x)) + ((z / x) * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) + Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x)) + Float64(Float64(z / x) * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024138 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))