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

Percentage Accurate: 94.2% → 98.5%

Time: 16.5s

Alternatives: 21

Speedup: 1.0×

• 31.4% 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: 94.2% 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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 + \frac{z \cdot t\_0 - -0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot t\_0 + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
   (if (<= x 1.2e+49)
     (+
      (- (* (+ x -0.5) (log x)) x)
      (+ 0.91893853320467 (/ (- (* z t_0) -0.083333333333333) x)))
     (+ (* (/ z x) t_0) (* x (+ (log x) -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651));
	double tmp;
	if (x <= 1.2e+49) {
		tmp = (((x + -0.5) * log(x)) - x) + (0.91893853320467 + (((z * t_0) - -0.083333333333333) / x));
	} else {
		tmp = ((z / x) * t_0) + (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) :: t_0
    real(8) :: tmp
    t_0 = (-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))
    if (x <= 1.2d+49) then
        tmp = (((x + (-0.5d0)) * log(x)) - x) + (0.91893853320467d0 + (((z * t_0) - (-0.083333333333333d0)) / x))
    else
        tmp = ((z / x) * t_0) + (x * (log(x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651));
	double tmp;
	if (x <= 1.2e+49) {
		tmp = (((x + -0.5) * Math.log(x)) - x) + (0.91893853320467 + (((z * t_0) - -0.083333333333333) / x));
	} else {
		tmp = ((z / x) * t_0) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651))
	tmp = 0
	if x <= 1.2e+49:
		tmp = (((x + -0.5) * math.log(x)) - x) + (0.91893853320467 + (((z * t_0) - -0.083333333333333) / x))
	else:
		tmp = ((z / x) * t_0) + (x * (math.log(x) + -1.0))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))
	tmp = 0.0
	if (x <= 1.2e+49)
		tmp = Float64(Float64(Float64(Float64(x + -0.5) * log(x)) - x) + Float64(0.91893853320467 + Float64(Float64(Float64(z * t_0) - -0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(z / x) * t_0) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651));
	tmp = 0.0;
	if (x <= 1.2e+49)
		tmp = (((x + -0.5) * log(x)) - x) + (0.91893853320467 + (((z * t_0) - -0.083333333333333) / x));
	else
		tmp = ((z / x) * t_0) + (x * (log(x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.2e+49], N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(z * t$95$0), $MachinePrecision] - -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e49

   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 99.8%

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

   if 1.2e49 < x

   1. Initial program 78.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 78.2%

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

      1. associate-+r- N/A

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

      2. div-sub N/A

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

      3. associate--r- N/A

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. log-lowering-log.f64 96.7%

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

   9. Simplified 96.7%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 + \frac{z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) - -0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. +-lowering-+.f64 N/A

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

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

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

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

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

   5. sub-neg N/A

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

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

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

   7. metadata-eval N/A

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

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

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

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

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

   12. distribute-neg-frac N/A

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

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

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

3. Simplified 91.2%

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

   1. associate-+r- N/A

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

   2. div-sub N/A

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

   3. associate--r- N/A

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

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

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

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

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

   6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

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

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

6. Applied egg-rr 98.5%

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

   1. associate--l- N/A

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

   2. associate-+l- N/A

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

   3. fmm-def N/A

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

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

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

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

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

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

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

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

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

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

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

   9. associate-+l- N/A

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

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

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

   11. sub-neg N/A

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

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

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

   13. distribute-neg-frac N/A

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

   14. /-lowering-/.f64 N/A

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

   15. metadata-eval N/A

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

   \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \frac{-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)}{\frac{x}{z}} - \left(x - \left(0.91893853320467 + \frac{0.083333333333333}{x}\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(t\_0 - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(-0.0027777777777778 + t\_0\right) + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ y 0.0007936500793651))))
   (if (<= x 2e+49)
     (+
      (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
      (/ (+ 0.083333333333333 (* z (- t_0 0.0027777777777778))) x))
     (+ (* (/ z x) (+ -0.0027777777777778 t_0)) (* x (+ (log x) -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y + 0.0007936500793651);
	double tmp;
	if (x <= 2e+49) {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * (t_0 - 0.0027777777777778))) / x);
	} else {
		tmp = ((z / x) * (-0.0027777777777778 + t_0)) + (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) :: t_0
    real(8) :: tmp
    t_0 = z * (y + 0.0007936500793651d0)
    if (x <= 2d+49) then
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + ((0.083333333333333d0 + (z * (t_0 - 0.0027777777777778d0))) / x)
    else
        tmp = ((z / x) * ((-0.0027777777777778d0) + t_0)) + (x * (log(x) + (-1.0d0)))
    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 <= 2e+49) {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * (t_0 - 0.0027777777777778))) / x);
	} else {
		tmp = ((z / x) * (-0.0027777777777778 + t_0)) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y + 0.0007936500793651)
	tmp = 0
	if x <= 2e+49:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * (t_0 - 0.0027777777777778))) / x)
	else:
		tmp = ((z / x) * (-0.0027777777777778 + t_0)) + (x * (math.log(x) + -1.0))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (x <= 2e+49)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(t_0 - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(Float64(z / x) * Float64(-0.0027777777777778 + t_0)) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y + 0.0007936500793651);
	tmp = 0.0;
	if (x <= 2e+49)
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * (t_0 - 0.0027777777777778))) / x);
	else
		tmp = ((z / x) * (-0.0027777777777778 + t_0)) + (x * (log(x) + -1.0));
	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, 2e+49], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(t$95$0 - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(-0.0027777777777778 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999989e49

   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. Add Preprocessing

   if 1.99999999999999989e49 < x

   1. Initial program 78.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 78.2%

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

      1. associate-+r- N/A

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

      2. div-sub N/A

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

      3. associate--r- N/A

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. log-lowering-log.f64 96.7%

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

   9. Simplified 96.7%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))
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) * (-0.0027777777777778 + (z * (y + 0.0007936500793651))));
}

Python

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

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(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x + -0.5) * log(x)) + (0.91893853320467 - x)) - (-0.083333333333333 / x)) + ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651))));
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[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 N/A

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

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

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

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

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

   5. sub-neg N/A

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

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

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

   7. metadata-eval N/A

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

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

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

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

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

   12. distribute-neg-frac N/A

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

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

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

3. Simplified 91.2%

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

   1. associate-+r- N/A

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

   2. div-sub N/A

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

   3. associate--r- N/A

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

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

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

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

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

   6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (((x * log(x)) + (0.91893853320467 - x)) - (-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 = ((z / x) * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0)))) + (((x * log(x)) + (0.91893853320467d0 - x)) - ((-0.083333333333333d0) / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. +-lowering-+.f64 N/A

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

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

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

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

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

   5. sub-neg N/A

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

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

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

   7. metadata-eval N/A

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

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

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

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

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

   12. distribute-neg-frac N/A

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

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

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

3. Simplified 91.2%

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

   1. associate-+r- N/A

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

   2. div-sub N/A

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

   3. associate--r- N/A

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

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

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

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

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

   6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

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

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

   6. log-lowering-log.f64 97.5%

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

9. Simplified 97.5%

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

10. Final simplification 97.5%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;x \leq 0.00225:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot t\_0 + x \cdot 0.91893853320467\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot t\_0 + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
   (if (<= x 0.00225)
     (/ (+ 0.083333333333333 (+ (* z t_0) (* x 0.91893853320467))) x)
     (+ (* (/ z x) t_0) (* x (+ (log x) -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651));
	double tmp;
	if (x <= 0.00225) {
		tmp = (0.083333333333333 + ((z * t_0) + (x * 0.91893853320467))) / x;
	} else {
		tmp = ((z / x) * t_0) + (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) :: t_0
    real(8) :: tmp
    t_0 = (-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))
    if (x <= 0.00225d0) then
        tmp = (0.083333333333333d0 + ((z * t_0) + (x * 0.91893853320467d0))) / x
    else
        tmp = ((z / x) * t_0) + (x * (log(x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651));
	double tmp;
	if (x <= 0.00225) {
		tmp = (0.083333333333333 + ((z * t_0) + (x * 0.91893853320467))) / x;
	} else {
		tmp = ((z / x) * t_0) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651))
	tmp = 0
	if x <= 0.00225:
		tmp = (0.083333333333333 + ((z * t_0) + (x * 0.91893853320467))) / x
	else:
		tmp = ((z / x) * t_0) + (x * (math.log(x) + -1.0))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))
	tmp = 0.0
	if (x <= 0.00225)
		tmp = Float64(Float64(0.083333333333333 + Float64(Float64(z * t_0) + Float64(x * 0.91893853320467))) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * t_0) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.0027777777777778 + (z * (y + 0.0007936500793651));
	tmp = 0.0;
	if (x <= 0.00225)
		tmp = (0.083333333333333 + ((z * t_0) + (x * 0.91893853320467))) / x;
	else
		tmp = ((z / x) * t_0) + (x * (log(x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.00225], N[(N[(0.083333333333333 + N[(N[(z * t$95$0), $MachinePrecision] + N[(x * 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00224999999999999983

   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 99.8%

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

      1. associate-+r- N/A

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

      2. div-sub N/A

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

      3. associate--r- N/A

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. +-commutative N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

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

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

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

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 99.2%

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

   12. Simplified 99.2%

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

   if 0.00224999999999999983 < x

   1. Initial program 81.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 81.7%

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

      1. associate-+r- N/A

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

      2. div-sub N/A

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

      3. associate--r- N/A

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. log-lowering-log.f64 95.6%

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

   9. Simplified 95.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00225:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot 0.91893853320467\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot 0.91893853320467\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8e+28)
   (/
    (+
     0.083333333333333
     (+
      (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651))))
      (* x 0.91893853320467)))
    x)
   (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+28) {
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * 0.91893853320467))) / x;
	} 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 <= 2.8d+28) then
        tmp = (0.083333333333333d0 + ((z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0)))) + (x * 0.91893853320467d0))) / x
    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 <= 2.8e+28) {
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * 0.91893853320467))) / x;
	} else {
		tmp = x * (Math.log(x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.8e+28:
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * 0.91893853320467))) / x
	else:
		tmp = x * (math.log(x) + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8e+28)
		tmp = Float64(Float64(0.083333333333333 + Float64(Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))) + Float64(x * 0.91893853320467))) / x);
	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 <= 2.8e+28)
		tmp = (0.083333333333333 + ((z * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (x * 0.91893853320467))) / x;
	else
		tmp = x * (log(x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.8e+28], N[(N[(0.083333333333333 + N[(N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e28

   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 99.8%

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

      1. associate-+r- N/A

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

      2. div-sub N/A

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

      3. associate--r- N/A

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

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

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

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

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

      6. associate-+l- N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. +-commutative N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

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

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

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

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 97.5%

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

   12. Simplified 97.5%

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

   if 2.8000000000000001e28 < x

   1. Initial program 79.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 79.5%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   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 71.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{log.f64}\left(x\right)\right)\right) \]

   7. Simplified 71.8%

      \[\leadsto \color{blue}{x \cdot \left(-1 + \log x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + x \cdot 0.91893853320467\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{\left(z \cdot y\right) \cdot \left(\frac{0.0007936500793651}{y} + 1\right)}{x}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (* (* z y) (+ (/ 0.0007936500793651 y) 1.0)) x))))
   (if (<= z -7.5e-25)
     t_0
     (if (<= z 1.9e-82) (* 0.083333333333333 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / x);
	double tmp;
	if (z <= -7.5e-25) {
		tmp = t_0;
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / 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 * y) * ((0.0007936500793651d0 / y) + 1.0d0)) / x)
    if (z <= (-7.5d-25)) then
        tmp = t_0
    else if (z <= 1.9d-82) then
        tmp = 0.083333333333333d0 * (1.0d0 / 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 * y) * ((0.0007936500793651 / y) + 1.0)) / x);
	double tmp;
	if (z <= -7.5e-25) {
		tmp = t_0;
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / x)
	tmp = 0
	if z <= -7.5e-25:
		tmp = t_0
	elif z <= 1.9e-82:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(Float64(z * y) * Float64(Float64(0.0007936500793651 / y) + 1.0)) / x))
	tmp = 0.0
	if (z <= -7.5e-25)
		tmp = t_0;
	elseif (z <= 1.9e-82)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / x);
	tmp = 0.0;
	if (z <= -7.5e-25)
		tmp = t_0;
	elseif (z <= 1.9e-82)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(N[(z * y), $MachinePrecision] * N[(N[(0.0007936500793651 / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.5e-25], t$95$0, If[LessEqual[z, 1.9e-82], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.49999999999999989e-25 or 1.9000000000000001e-82 < z

   1. Initial program 86.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 60.5%

      \[\leadsto \color{blue}{y \cdot \left(\log x \cdot \frac{x + -0.5}{y} + \left(\frac{0.91893853320467}{y} - \left(\left(0 - \frac{z \cdot z}{x}\right) + \frac{x + \left(\frac{-0.083333333333333}{x} - \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right)}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 68.7%

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

   10. Simplified 68.7%

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

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{\left(y \cdot {z}^{2}\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x} \]

       2. associate-/l* N/A

          \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}} \]

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

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

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

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

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

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. /-lowering-/.f64 72.6%

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

   13. Simplified 72.6%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{1 + \frac{0.0007936500793651}{y}}{x}} \]
   14. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

       9. /-lowering-/.f64 75.5%

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

   15. Applied egg-rr 75.5%

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

   if -7.49999999999999989e-25 < z < 1.9000000000000001e-82

   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 53.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 53.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \frac{\left(z \cdot y\right) \cdot \left(\frac{0.0007936500793651}{y} + 1\right)}{x}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(z \cdot y\right) \cdot \left(\frac{0.0007936500793651}{y} + 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(z \cdot \frac{\frac{0.0007936500793651}{y} + 1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e-25)
   (* (+ y 0.0007936500793651) (/ (* z z) x))
   (if (<= z 1.9e-82)
     (* 0.083333333333333 (/ 1.0 x))
     (* (* z y) (* z (/ (+ (/ 0.0007936500793651 y) 1.0) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-25) {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = (z * y) * (z * (((0.0007936500793651 / y) + 1.0) / 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) :: tmp
    if (z <= (-4.8d-25)) then
        tmp = (y + 0.0007936500793651d0) * ((z * z) / x)
    else if (z <= 1.9d-82) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = (z * y) * (z * (((0.0007936500793651d0 / y) + 1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-25) {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = (z * y) * (z * (((0.0007936500793651 / y) + 1.0) / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e-25:
		tmp = (y + 0.0007936500793651) * ((z * z) / x)
	elif z <= 1.9e-82:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = (z * y) * (z * (((0.0007936500793651 / y) + 1.0) / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e-25)
		tmp = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x));
	elseif (z <= 1.9e-82)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(Float64(z * y) * Float64(z * Float64(Float64(Float64(0.0007936500793651 / y) + 1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e-25)
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	elseif (z <= 1.9e-82)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = (z * y) * (z * (((0.0007936500793651 / y) + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e-25], N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e-82], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(z * N[(N[(N[(0.0007936500793651 / y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000018e-25

   1. Initial program 86.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 86.9%

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

   5. Taylor expanded in z around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}{x} \]

      2. associate-/l* N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right), \mathsf{/.f64}\left(\left({z}^{2}\right), \color{blue}{x}\right)\right) \]

      6. unpow2 N/A

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

      7. *-lowering-*.f64 81.6%

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

   7. Simplified 81.6%

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

   if -4.80000000000000018e-25 < z < 1.9000000000000001e-82

   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 53.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 53.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 1.9000000000000001e-82 < z

   1. Initial program 85.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 85.7%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 62.8%

      \[\leadsto \color{blue}{y \cdot \left(\log x \cdot \frac{x + -0.5}{y} + \left(\frac{0.91893853320467}{y} - \left(\left(0 - \frac{z \cdot z}{x}\right) + \frac{x + \left(\frac{-0.083333333333333}{x} - \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right)}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 60.5%

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

   10. Simplified 60.5%

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

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{\left(y \cdot {z}^{2}\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x} \]

       2. associate-/l* N/A

          \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}} \]

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

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

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

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

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

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

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

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. /-lowering-/.f64 64.1%

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

   13. Simplified 64.1%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{1 + \frac{0.0007936500793651}{y}}{x}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 64.3%

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

   15. Applied egg-rr 64.3%

       \[\leadsto \color{blue}{\left(\frac{1 + \frac{0.0007936500793651}{y}}{x} \cdot z\right) \cdot \left(z \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(z \cdot \frac{\frac{0.0007936500793651}{y} + 1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-24}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 10^{-83}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0007936500793651}{y} + 1}{x} \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.05e-24)
   (* (+ y 0.0007936500793651) (/ (* z z) x))
   (if (<= z 1e-83)
     (* 0.083333333333333 (/ 1.0 x))
     (* (/ (+ (/ 0.0007936500793651 y) 1.0) x) (* z (* z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e-24) {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	} else if (z <= 1e-83) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = (((0.0007936500793651 / y) + 1.0) / x) * (z * (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 (z <= (-1.05d-24)) then
        tmp = (y + 0.0007936500793651d0) * ((z * z) / x)
    else if (z <= 1d-83) then
        tmp = 0.083333333333333d0 * (1.0d0 / x)
    else
        tmp = (((0.0007936500793651d0 / y) + 1.0d0) / x) * (z * (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e-24) {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	} else if (z <= 1e-83) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = (((0.0007936500793651 / y) + 1.0) / x) * (z * (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.05e-24:
		tmp = (y + 0.0007936500793651) * ((z * z) / x)
	elif z <= 1e-83:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = (((0.0007936500793651 / y) + 1.0) / x) * (z * (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.05e-24)
		tmp = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x));
	elseif (z <= 1e-83)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 / y) + 1.0) / x) * Float64(z * Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.05e-24)
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	elseif (z <= 1e-83)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = (((0.0007936500793651 / y) + 1.0) / x) * (z * (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.05e-24], N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-83], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 / y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05e-24

   1. Initial program 86.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 86.9%

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

   5. Taylor expanded in z around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}{x} \]

      2. associate-/l* N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right), \mathsf{/.f64}\left(\left({z}^{2}\right), \color{blue}{x}\right)\right) \]

      6. unpow2 N/A

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

      7. *-lowering-*.f64 81.6%

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

   7. Simplified 81.6%

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

   if -1.05e-24 < z < 1e-83

   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. +-lowering-+.f64 N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

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

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

      12. distribute-neg-frac N/A

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

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

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 98.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 53.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 53.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

   if 1e-83 < z

   1. Initial program 85.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \left(-1 \cdot \frac{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}}{y} + \left(-1 \cdot \frac{{z}^{2}}{x} + \frac{x}{y}\right)\right)\right)} \]

   7. Simplified 62.8%

      \[\leadsto \color{blue}{y \cdot \left(\log x \cdot \frac{x + -0.5}{y} + \left(\frac{0.91893853320467}{y} - \left(\left(0 - \frac{z \cdot z}{x}\right) + \frac{x + \left(\frac{-0.083333333333333}{x} - \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right)}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({z}^{2}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \frac{1}{x}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(z \cdot z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{1}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} \cdot \frac{1}{x \cdot y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x \cdot y}}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x} \cdot y}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 60.5%

       \[\leadsto y \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{y \cdot x}\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{y \cdot \left({z}^{2} \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}{x}} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(y \cdot {z}^{2}\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x} \]

       2. associate-/l* N/A

          \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{\left(\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({z}^{2} \cdot y\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(z \cdot z\right) \cdot y\right), \left(\frac{\color{blue}{1} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot y\right)\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(y \cdot z\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(y \cdot z\right)\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot y\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right), \color{blue}{x}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right), x\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{y}\right)\right), x\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right), x\right)\right) \]

       15. /-lowering-/.f64 64.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   13. Simplified 64.1%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{1 + \frac{0.0007936500793651}{y}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-24}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 10^{-83}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0007936500793651}{y} + 1}{x} \cdot \left(z \cdot \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{x \cdot y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.5e+76)
   (/
    (+
     0.083333333333333
     (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
    x)
   (* z (* y (* z (+ (/ 1.0 x) (/ 0.0007936500793651 (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+76) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = z * (y * (z * ((1.0 / x) + (0.0007936500793651 / (x * 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 <= 4.5d+76) then
        tmp = (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x
    else
        tmp = z * (y * (z * ((1.0d0 / x) + (0.0007936500793651d0 / (x * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+76) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = z * (y * (z * ((1.0 / x) + (0.0007936500793651 / (x * y)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.5e+76:
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x
	else:
		tmp = z * (y * (z * ((1.0 / x) + (0.0007936500793651 / (x * y)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.5e+76)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x);
	else
		tmp = Float64(z * Float64(y * Float64(z * Float64(Float64(1.0 / x) + Float64(0.0007936500793651 / Float64(x * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.5e+76)
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	else
		tmp = z * (y * (z * ((1.0 / x) + (0.0007936500793651 / (x * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.5e+76], N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(y * N[(z * N[(N[(1.0 / x), $MachinePrecision] + N[(0.0007936500793651 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999997e76

   1. Initial program 99.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   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. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right)}{x} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(-1 \cdot \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-1 \cdot \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)\right), \color{blue}{x}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right)\right), x\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      12. +-lowering-+.f64 89.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

   7. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right)}{x}} \]

   if 4.4999999999999997e76 < x

   1. Initial program 75.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \left(-1 \cdot \frac{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}}{y} + \left(-1 \cdot \frac{{z}^{2}}{x} + \frac{x}{y}\right)\right)\right)} \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{y \cdot \left(\log x \cdot \frac{x + -0.5}{y} + \left(\frac{0.91893853320467}{y} - \left(\left(0 - \frac{z \cdot z}{x}\right) + \frac{x + \left(\frac{-0.083333333333333}{x} - \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right)}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({z}^{2}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \frac{1}{x}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(z \cdot z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{1}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} \cdot \frac{1}{x \cdot y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x \cdot y}}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x} \cdot y}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 22.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 22.0%

       \[\leadsto y \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{y \cdot x}\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right) \cdot \color{blue}{y} \]

       2. associate-*l* N/A

          \[\leadsto \left(z \cdot \left(z \cdot \left(\frac{1}{x} + \frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right)\right) \cdot y \]

       3. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(z \cdot \left(\frac{1}{x} + \frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right) \cdot y\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\left(z \cdot \left(\frac{1}{x} + \frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right) \cdot y\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\left(z \cdot \left(\frac{1}{x} + \frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right), \color{blue}{y}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{1}{x} + \frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right), y\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \left(\frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right)\right), y\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000}}{y \cdot x}\right)\right)\right), y\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \left(y \cdot x\right)\right)\right)\right), y\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \left(x \cdot y\right)\right)\right)\right), y\right)\right) \]

       11. *-lowering-*.f64 28.3%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right), y\right)\right) \]

   12. Applied egg-rr 28.3%

       \[\leadsto \color{blue}{z \cdot \left(\left(z \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{x \cdot y}\right)\right) \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(z \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{x \cdot y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.0007936500793651) (/ (* z z) x))))
   (if (<= z -3.8e-25)
     t_0
     (if (<= z 1.9e-82) (* 0.083333333333333 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * ((z * z) / x);
	double tmp;
	if (z <= -3.8e-25) {
		tmp = t_0;
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / 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 = (y + 0.0007936500793651d0) * ((z * z) / x)
    if (z <= (-3.8d-25)) then
        tmp = t_0
    else if (z <= 1.9d-82) then
        tmp = 0.083333333333333d0 * (1.0d0 / 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 = (y + 0.0007936500793651) * ((z * z) / x);
	double tmp;
	if (z <= -3.8e-25) {
		tmp = t_0;
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + 0.0007936500793651) * ((z * z) / x)
	tmp = 0
	if z <= -3.8e-25:
		tmp = t_0
	elif z <= 1.9e-82:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x))
	tmp = 0.0
	if (z <= -3.8e-25)
		tmp = t_0;
	elseif (z <= 1.9e-82)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + 0.0007936500793651) * ((z * z) / x);
	tmp = 0.0;
	if (z <= -3.8e-25)
		tmp = t_0;
	elseif (z <= 1.9e-82)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e-25], t$95$0, If[LessEqual[z, 1.9e-82], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999998e-25 or 1.9000000000000001e-82 < z

   1. Initial program 86.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \color{blue}{\frac{{z}^{2}}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{7936500793651}{10000000000000000} + y\right), \color{blue}{\left(\frac{{z}^{2}}{x}\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right), \left(\frac{\color{blue}{{z}^{2}}}{x}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right), \mathsf{/.f64}\left(\left({z}^{2}\right), \color{blue}{x}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right), \mathsf{/.f64}\left(\left(z \cdot z\right), x\right)\right) \]

      7. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), x\right)\right) \]

   7. Simplified 72.8%

      \[\leadsto \color{blue}{\left(0.0007936500793651 + y\right) \cdot \frac{z \cdot z}{x}} \]

   if -3.7999999999999998e-25 < z < 1.9000000000000001e-82

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 98.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 53.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 53.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 53.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-25}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(z \cdot y\right) \cdot \left(\frac{0.0007936500793651}{y} + 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.5e+53)
   (/
    (+
     0.083333333333333
     (* z (+ -0.0027777777777778 (* z (+ y 0.0007936500793651)))))
    x)
   (* z (/ (* (* z y) (+ (/ 0.0007936500793651 y) 1.0)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.5e+53) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / 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) :: tmp
    if (x <= 6.5d+53) then
        tmp = (0.083333333333333d0 + (z * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))) / x
    else
        tmp = z * (((z * y) * ((0.0007936500793651d0 / y) + 1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.5e+53) {
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	} else {
		tmp = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.5e+53:
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x
	else:
		tmp = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.5e+53)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651))))) / x);
	else
		tmp = Float64(z * Float64(Float64(Float64(z * y) * Float64(Float64(0.0007936500793651 / y) + 1.0)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6.5e+53)
		tmp = (0.083333333333333 + (z * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))) / x;
	else
		tmp = z * (((z * y) * ((0.0007936500793651 / y) + 1.0)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.5e+53], N[(N[(0.083333333333333 + N[(z * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(N[(N[(z * y), $MachinePrecision] * N[(N[(0.0007936500793651 / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000017e53

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   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. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right)}{x} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(-1 \cdot \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)}{x} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-1 \cdot \left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)\right), \color{blue}{x}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right)\right), x\right) \]

      5. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} + y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

      12. +-lowering-+.f64 93.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{83333333333333}{1000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), \frac{-13888888888889}{5000000000000000}\right)\right)\right), x\right) \]

   7. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) + -0.0027777777777778\right)}{x}} \]

   if 6.50000000000000017e53 < x

   1. Initial program 77.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \left(-1 \cdot \frac{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}}{y} + \left(-1 \cdot \frac{{z}^{2}}{x} + \frac{x}{y}\right)\right)\right)} \]

   7. Simplified 62.3%

      \[\leadsto \color{blue}{y \cdot \left(\log x \cdot \frac{x + -0.5}{y} + \left(\frac{0.91893853320467}{y} - \left(\left(0 - \frac{z \cdot z}{x}\right) + \frac{x + \left(\frac{-0.083333333333333}{x} - \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right)}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({z}^{2}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \frac{1}{x}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(z \cdot z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{1}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} \cdot \frac{1}{x \cdot y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x \cdot y}}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x} \cdot y}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 22.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 22.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{y \cdot x}\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{y \cdot \left({z}^{2} \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}{x}} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(y \cdot {z}^{2}\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x} \]

       2. associate-/l* N/A

          \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{\left(\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({z}^{2} \cdot y\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(z \cdot z\right) \cdot y\right), \left(\frac{\color{blue}{1} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot y\right)\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(y \cdot z\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(y \cdot z\right)\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot y\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right), \color{blue}{x}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right), x\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{y}\right)\right), x\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right), x\right)\right) \]

       15. /-lowering-/.f64 25.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   13. Simplified 25.2%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{1 + \frac{0.0007936500793651}{y}}{x}} \]
   14. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(\left(z \cdot y\right) \cdot \frac{1 + \frac{\frac{7936500793651}{10000000000000000}}{y}}{x}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(z \cdot y\right) \cdot \frac{1 + \frac{\frac{7936500793651}{10000000000000000}}{y}}{x}\right) \cdot \color{blue}{z} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(z \cdot y\right) \cdot \frac{1 + \frac{\frac{7936500793651}{10000000000000000}}{y}}{x}\right), \color{blue}{z}\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(z \cdot y\right) \cdot \left(1 + \frac{\frac{7936500793651}{10000000000000000}}{y}\right)}{x}\right), z\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(z \cdot y\right) \cdot \left(1 + \frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right), x\right), z\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(z \cdot y\right), \left(1 + \frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right), x\right), z\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \left(1 + \frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right), x\right), z\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right)\right), x\right), z\right) \]

       9. /-lowering-/.f64 28.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, y\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right)\right), x\right), z\right) \]

   15. Applied egg-rr 28.0%

       \[\leadsto \color{blue}{\frac{\left(z \cdot y\right) \cdot \left(1 + \frac{0.0007936500793651}{y}\right)}{x} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\left(z \cdot y\right) \cdot \left(\frac{0.0007936500793651}{y} + 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;z \leq -6600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-83}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (* z z) x))))
   (if (<= z -6600000000.0)
     t_0
     (if (<= z 6.4e-83) (* 0.083333333333333 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((z * z) / x);
	double tmp;
	if (z <= -6600000000.0) {
		tmp = t_0;
	} else if (z <= 6.4e-83) {
		tmp = 0.083333333333333 * (1.0 / 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 = y * ((z * z) / x)
    if (z <= (-6600000000.0d0)) then
        tmp = t_0
    else if (z <= 6.4d-83) then
        tmp = 0.083333333333333d0 * (1.0d0 / 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 = y * ((z * z) / x);
	double tmp;
	if (z <= -6600000000.0) {
		tmp = t_0;
	} else if (z <= 6.4e-83) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((z * z) / x)
	tmp = 0
	if z <= -6600000000.0:
		tmp = t_0
	elif z <= 6.4e-83:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(z * z) / x))
	tmp = 0.0
	if (z <= -6600000000.0)
		tmp = t_0;
	elseif (z <= 6.4e-83)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((z * z) / x);
	tmp = 0.0;
	if (z <= -6600000000.0)
		tmp = t_0;
	elseif (z <= 6.4e-83)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6600000000.0], t$95$0, If[LessEqual[z, 6.4e-83], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e9 or 6.4000000000000002e-83 < z

   1. Initial program 86.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   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 57.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]

      2. +-lft-identity N/A

         \[\leadsto y \cdot \left(0 + \color{blue}{\frac{z \cdot z}{x}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(0 + \frac{z \cdot z}{x}\right) \cdot \color{blue}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 + \frac{z \cdot z}{x}\right), \color{blue}{y}\right) \]

      5. +-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{z \cdot z}{x}\right), y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(z \cdot z\right), x\right), y\right) \]

      7. *-lowering-*.f64 61.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), x\right), y\right) \]

   9. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{z \cdot z}{x} \cdot y} \]

   if -6.6e9 < z < 6.4000000000000002e-83

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 96.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 96.6%

      \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 52.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 52.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 52.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6600000000:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-83}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \frac{y}{x}\\ \mathbf{if}\;z \leq -65000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-44}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) (/ y x))))
   (if (<= z -65000000000000.0)
     t_0
     (if (<= z 3.6e-44) (* 0.083333333333333 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * z) * (y / x);
	double tmp;
	if (z <= -65000000000000.0) {
		tmp = t_0;
	} else if (z <= 3.6e-44) {
		tmp = 0.083333333333333 * (1.0 / 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) * (y / x)
    if (z <= (-65000000000000.0d0)) then
        tmp = t_0
    else if (z <= 3.6d-44) then
        tmp = 0.083333333333333d0 * (1.0d0 / 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) * (y / x);
	double tmp;
	if (z <= -65000000000000.0) {
		tmp = t_0;
	} else if (z <= 3.6e-44) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * z) * (y / x)
	tmp = 0
	if z <= -65000000000000.0:
		tmp = t_0
	elif z <= 3.6e-44:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * z) * Float64(y / x))
	tmp = 0.0
	if (z <= -65000000000000.0)
		tmp = t_0;
	elseif (z <= 3.6e-44)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * z) * (y / x);
	tmp = 0.0;
	if (z <= -65000000000000.0)
		tmp = t_0;
	elseif (z <= 3.6e-44)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -65000000000000.0], t$95$0, If[LessEqual[z, 3.6e-44], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e13 or 3.5999999999999999e-44 < z

   1. Initial program 85.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   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 59.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(z \cdot z\right) \cdot y}{x} \]

      2. associate-/l* N/A

         \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\frac{y}{x}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

      5. /-lowering-/.f64 60.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot \frac{y}{x}} \]

   if -6.5e13 < z < 3.5999999999999999e-44

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 95.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 95.0%

      \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 50.1%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 50.1%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 50.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 50.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -65000000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y}{x}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-44}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{if}\;z \leq -6600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z x) (* z y))))
   (if (<= z -6600000000.0)
     t_0
     (if (<= z 1.9e-82) (* 0.083333333333333 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z / x) * (z * y);
	double tmp;
	if (z <= -6600000000.0) {
		tmp = t_0;
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / 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 (z <= (-6600000000.0d0)) then
        tmp = t_0
    else if (z <= 1.9d-82) then
        tmp = 0.083333333333333d0 * (1.0d0 / 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 (z <= -6600000000.0) {
		tmp = t_0;
	} else if (z <= 1.9e-82) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / x) * (z * y)
	tmp = 0
	if z <= -6600000000.0:
		tmp = t_0
	elif z <= 1.9e-82:
		tmp = 0.083333333333333 * (1.0 / 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 (z <= -6600000000.0)
		tmp = t_0;
	elseif (z <= 1.9e-82)
		tmp = Float64(0.083333333333333 * Float64(1.0 / 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 (z <= -6600000000.0)
		tmp = t_0;
	elseif (z <= 1.9e-82)
		tmp = 0.083333333333333 * (1.0 / 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[z, -6600000000.0], t$95$0, If[LessEqual[z, 1.9e-82], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e9 or 1.9000000000000001e-82 < z

   1. Initial program 86.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   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 57.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

   7. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   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 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 -6.6e9 < z < 1.9000000000000001e-82

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 96.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 96.6%

      \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 52.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 52.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 52.1%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6600000000:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{if}\;z \leq -10.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* 0.0007936500793651 (/ z x)))))
   (if (<= z -10.2)
     t_0
     (if (<= z 1.25e-13) (* 0.083333333333333 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (0.0007936500793651 * (z / x));
	double tmp;
	if (z <= -10.2) {
		tmp = t_0;
	} else if (z <= 1.25e-13) {
		tmp = 0.083333333333333 * (1.0 / 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 * (0.0007936500793651d0 * (z / x))
    if (z <= (-10.2d0)) then
        tmp = t_0
    else if (z <= 1.25d-13) then
        tmp = 0.083333333333333d0 * (1.0d0 / 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 * (0.0007936500793651 * (z / x));
	double tmp;
	if (z <= -10.2) {
		tmp = t_0;
	} else if (z <= 1.25e-13) {
		tmp = 0.083333333333333 * (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (0.0007936500793651 * (z / x))
	tmp = 0
	if z <= -10.2:
		tmp = t_0
	elif z <= 1.25e-13:
		tmp = 0.083333333333333 * (1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(0.0007936500793651 * Float64(z / x)))
	tmp = 0.0
	if (z <= -10.2)
		tmp = t_0;
	elseif (z <= 1.25e-13)
		tmp = Float64(0.083333333333333 * Float64(1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (0.0007936500793651 * (z / x));
	tmp = 0.0;
	if (z <= -10.2)
		tmp = t_0;
	elseif (z <= 1.25e-13)
		tmp = 0.083333333333333 * (1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(0.0007936500793651 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -10.2], t$95$0, If[LessEqual[z, 1.25e-13], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -10.199999999999999 or 1.24999999999999997e-13 < z

   1. Initial program 85.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{91893853320467}{100000000000000} \cdot \frac{1}{y} + \frac{\log x \cdot \left(x - \frac{1}{2}\right)}{y}\right) - \left(-1 \cdot \frac{\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}}{y} + \left(-1 \cdot \frac{{z}^{2}}{x} + \frac{x}{y}\right)\right)\right)} \]

   7. Simplified 58.5%

      \[\leadsto \color{blue}{y \cdot \left(\log x \cdot \frac{x + -0.5}{y} + \left(\frac{0.91893853320467}{y} - \left(\left(0 - \frac{z \cdot z}{x}\right) + \frac{x + \left(\frac{-0.083333333333333}{x} - \frac{z \cdot \left(-0.0027777777777778 + z \cdot 0.0007936500793651\right)}{x}\right)}{y}\right)\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left({z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({z}^{2}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y} + \frac{1}{x}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(z \cdot z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}} + \frac{1}{x}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{1}{x} + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right)}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{7936500793651}{10000000000000000}} \cdot \frac{1}{x \cdot y}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x \cdot y}}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x} \cdot y}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 71.2%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 71.2%

       \[\leadsto y \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{0.0007936500793651}{y \cdot x}\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{y \cdot \left({z}^{2} \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right)}{x}} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(y \cdot {z}^{2}\right) \cdot \left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)}{x} \]

       2. associate-/l* N/A

          \[\leadsto \left(y \cdot {z}^{2}\right) \cdot \color{blue}{\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {z}^{2}\right), \color{blue}{\left(\frac{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({z}^{2} \cdot y\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(z \cdot z\right) \cdot y\right), \left(\frac{\color{blue}{1} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}{x}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot y\right)\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot \left(y \cdot z\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(y \cdot z\right)\right), \left(\frac{\color{blue}{1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot y\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \left(\frac{1 + \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}}}{x}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right), \color{blue}{x}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{y}\right)\right), x\right)\right) \]

       13. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{y}\right)\right), x\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{7936500793651}{10000000000000000}}{y}\right)\right), x\right)\right) \]

       15. /-lowering-/.f64 75.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, y\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{7936500793651}{10000000000000000}, y\right)\right), x\right)\right) \]

   13. Simplified 75.4%

       \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{1 + \frac{0.0007936500793651}{y}}{x}} \]

   14. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} \]
   15. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{z}^{2}}{x} \cdot \color{blue}{\frac{7936500793651}{10000000000000000}} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]

       3. associate-*r/ N/A

          \[\leadsto {z}^{2} \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} \]

       4. metadata-eval N/A

          \[\leadsto {z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x} \]

       5. associate-*r/ N/A

          \[\leadsto {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{1}{x}}\right) \]

       6. unpow2 N/A

          \[\leadsto \left(z \cdot z\right) \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000}} \cdot \frac{1}{x}\right) \]

       7. associate-*l* N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)\right)}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} \cdot 1}{\color{blue}{x}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{7936500793651}{10000000000000000}}{x}\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{7936500793651}{10000000000000000}}{\color{blue}{x}}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x}\right)\right) \]

       13. *-lft-identity N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000} \cdot z}{1 \cdot \color{blue}{x}}\right)\right) \]

       14. times-frac N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{7936500793651}{10000000000000000}}{1} \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{7936500793651}{10000000000000000} \cdot \frac{\color{blue}{z}}{x}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\left(\frac{z}{x}\right)}\right)\right) \]

       17. /-lowering-/.f64 50.0%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\frac{7936500793651}{10000000000000000}, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   16. Simplified 50.0%

       \[\leadsto \color{blue}{z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)} \]

   if -10.199999999999999 < z < 1.24999999999999997e-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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 94.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 94.2%

      \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 48.7%

         \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

   10. Simplified 48.7%

       \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

       4. /-lowering-/.f64 48.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

   12. Applied egg-rr 48.8%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10.2:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + \left(0.91893853320467 - \frac{-0.083333333333333}{x}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (* (/ z x) (+ -0.0027777777777778 (* z (+ y 0.0007936500793651))))
  (- 0.91893853320467 (/ -0.083333333333333 x))))

C

double code(double x, double y, double z) {
	return ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (0.91893853320467 - (-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 = ((z / x) * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0)))) + (0.91893853320467d0 - ((-0.083333333333333d0) / x))
end function

Java

public static double code(double x, double y, double z) {
	return ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (0.91893853320467 - (-0.083333333333333 / x));
}

Python

def code(x, y, z):
	return ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (0.91893853320467 - (-0.083333333333333 / x))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))) + Float64(0.91893853320467 - Float64(-0.083333333333333 / x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651)))) + (0.91893853320467 - (-0.083333333333333 / x));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z / x), $MachinePrecision] * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r- N/A

      \[\leadsto \left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\frac{\frac{-83333333333333}{1000000000000000} - z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}} \]

   2. div-sub N/A

      \[\leadsto \left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \left(\frac{\frac{-83333333333333}{1000000000000000}}{x} - \color{blue}{\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}}\right) \]

   3. associate--r- N/A

      \[\leadsto \left(\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \frac{\frac{-83333333333333}{1000000000000000}}{x}\right) + \color{blue}{\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \frac{\frac{-83333333333333}{1000000000000000}}{x}\right), \color{blue}{\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}}{x}\right)\right) \]

   6. associate-+l- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{\color{blue}{z} \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{\color{blue}{z} \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \frac{-1}{2}\right), \log x\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   10. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \left(\frac{z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}}{x}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x}\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right), \color{blue}{\left(\frac{z}{x}\right)}\right)\right) \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(\left(\left(x + -0.5\right) \cdot \log x - \left(x - 0.91893853320467\right)\right) - \frac{-0.083333333333333}{x}\right) + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x}} \]

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)\right)}, \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{x}\right)\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{z}, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{z}, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]

   3. log-rec N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]

   4. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \log x\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log x\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{z}, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]

   6. log-lowering-log.f64 97.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]

9. Simplified 97.5%

   \[\leadsto \left(\left(\color{blue}{x \cdot \log x} - \left(x - 0.91893853320467\right)\right) - \frac{-0.083333333333333}{x}\right) + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\frac{91893853320467}{100000000000000}}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]
11. Step-by-step derivation

12. Simplified 68.7%

    \[\leadsto \left(\color{blue}{0.91893853320467} - \frac{-0.083333333333333}{x}\right) + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x} \]

13. Final simplification 68.7%

    \[\leadsto \frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) + \left(0.91893853320467 - \frac{-0.083333333333333}{x}\right) \]
14. Add Preprocessing

Alternative 19: 64.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} + \frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (/ 0.083333333333333 x)
  (* (/ z x) (+ -0.0027777777777778 (* z (+ y 0.0007936500793651))))))

C

double code(double x, double y, double z) {
	return (0.083333333333333 / x) + ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651))));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (0.083333333333333d0 / x) + ((z / x) * ((-0.0027777777777778d0) + (z * (y + 0.0007936500793651d0))))
end function

Java

public static double code(double x, double y, double z) {
	return (0.083333333333333 / x) + ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651))));
}

Python

def code(x, y, z):
	return (0.083333333333333 / x) + ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651))))

Julia

function code(x, y, z)
	return Float64(Float64(0.083333333333333 / x) + Float64(Float64(z / x) * Float64(-0.0027777777777778 + Float64(z * Float64(y + 0.0007936500793651)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.083333333333333 / x) + ((z / x) * (-0.0027777777777778 + (z * (y + 0.0007936500793651))));
end

Wolfram

code[x_, y_, z_] := N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(-0.0027777777777778 + N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r- N/A

      \[\leadsto \left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \color{blue}{\frac{\frac{-83333333333333}{1000000000000000} - z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}} \]

   2. div-sub N/A

      \[\leadsto \left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \left(\frac{\frac{-83333333333333}{1000000000000000}}{x} - \color{blue}{\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}}\right) \]

   3. associate--r- N/A

      \[\leadsto \left(\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \frac{\frac{-83333333333333}{1000000000000000}}{x}\right) + \color{blue}{\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) - \frac{\frac{-83333333333333}{1000000000000000}}{x}\right), \color{blue}{\left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)}\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}}{x}\right)\right) \]

   6. associate-+l- N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{\color{blue}{z} \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\left(x + \frac{-1}{2}\right) \cdot \log x\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{\color{blue}{z} \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \frac{-1}{2}\right), \log x\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   10. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \left(x - \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)\right), \left(\frac{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}{x}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \left(\frac{z \cdot \color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right)}}{x}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) \cdot z}{x}\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right) \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), \mathsf{\_.f64}\left(x, \frac{91893853320467}{100000000000000}\right)\right), \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, x\right)\right), \mathsf{*.f64}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \frac{-13888888888889}{5000000000000000}\right), \color{blue}{\left(\frac{z}{x}\right)}\right)\right) \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(\left(\left(x + -0.5\right) \cdot \log x - \left(x - 0.91893853320467\right)\right) - \frac{-0.083333333333333}{x}\right) + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x}\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right), \mathsf{/.f64}\left(z, x\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 68.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, x\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, \frac{7936500793651}{10000000000000000}\right)\right), \frac{-13888888888889}{5000000000000000}\right)}, \mathsf{/.f64}\left(z, x\right)\right)\right) \]

9. Simplified 68.3%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} + \left(z \cdot \left(y + 0.0007936500793651\right) + -0.0027777777777778\right) \cdot \frac{z}{x} \]

10. Final simplification 68.3%

    \[\leadsto \frac{0.083333333333333}{x} + \frac{z}{x} \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right) \]
11. Add Preprocessing

Alternative 20: 23.1% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ 0.083333333333333 \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.083333333333333 (/ 1.0 x)))

C

double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / 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 = 0.083333333333333d0 * (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Python

def code(x, y, z):
	return 0.083333333333333 * (1.0 / x)

Julia

function code(x, y, z)
	return Float64(0.083333333333333 * Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 * (1.0 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 53.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

7. Simplified 53.3%

   \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 22.8%

      \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

10. Simplified 22.8%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

    2. associate-/r/ N/A

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{83333333333333}{1000000000000000}}\right) \]

    4. /-lowering-/.f64 22.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{83333333333333}{1000000000000000}\right) \]

12. Applied egg-rr 22.8%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

13. Final simplification 22.8%

    \[\leadsto 0.083333333333333 \cdot \frac{1}{x} \]
14. Add Preprocessing

Alternative 21: 23.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 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 = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right), \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x\right), x\right), \left(\color{blue}{\frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x - \frac{1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \log x\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \left(\frac{91893853320467}{100000000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)}\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \left(\frac{\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)}{\color{blue}{x}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \frac{-0.083333333333333 - z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \color{blue}{\left(\frac{\frac{-83333333333333}{1000000000000000}}{x}\right)}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 53.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{log.f64}\left(x\right)\right), x\right), \mathsf{\_.f64}\left(\frac{91893853320467}{100000000000000}, \mathsf{/.f64}\left(\frac{-83333333333333}{1000000000000000}, \color{blue}{x}\right)\right)\right) \]

7. Simplified 53.3%

   \[\leadsto \left(\left(x + -0.5\right) \cdot \log x - x\right) + \left(0.91893853320467 - \color{blue}{\frac{-0.083333333333333}{x}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 22.8%

      \[\leadsto \mathsf{/.f64}\left(\frac{83333333333333}{1000000000000000}, \color{blue}{x}\right) \]

10. Simplified 22.8%

    \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
11. Add Preprocessing

Developer Target 1: 98.6% 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 2024160 
(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)))