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

Percentage Accurate: 93.5% → 99.1%

Time: 16.7s

Alternatives: 21

Speedup: 1.0×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - \frac{0.5}{x}\right) \cdot x, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(\frac{y}{x} \cdot z, z, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.4e+23)
   (fma
    (* (- 1.0 (/ 0.5 x)) x)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (fma
    (- x 0.5)
    (log x)
    (+
     (fma
      (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
      z
      (fma (* (/ y x) z) z (/ 0.083333333333333 x)))
     (- 0.91893853320467 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.4e+23) {
		tmp = fma(((1.0 - (0.5 / x)) * x), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = fma((x - 0.5), log(x), (fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, fma(((y / x) * z), z, (0.083333333333333 / x))) + (0.91893853320467 - x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.4e+23)
		tmp = fma(Float64(Float64(1.0 - Float64(0.5 / x)) * x), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, fma(Float64(Float64(y / x) * z), z, Float64(0.083333333333333 / x))) + Float64(0.91893853320467 - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.4e+23], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.40000000000000017e23

   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
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.7

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 4.40000000000000017e23 < x

   1. Initial program 86.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 86.5

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

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 86.5

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 86.5

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 86.5%

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

   7. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-*l/ N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - \frac{0.5}{x}\right) \cdot x, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \mathsf{fma}\left(\frac{y}{x} \cdot z, z, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+282}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(\left(0.0007936500793651 + y\right) - \frac{0.0027777777777778}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))
   (if (<= t_0 -1e+282)
     (* (* (/ y x) z) z)
     (if (<= t_0 5e+305)
       (fma
        (- x 0.5)
        (log x)
        (+
         (/
          (fma
           (fma 0.0007936500793651 z -0.0027777777777778)
           z
           0.083333333333333)
          x)
         (- 0.91893853320467 x)))
       (*
        (* (/ z x) z)
        (- (+ 0.0007936500793651 y) (/ 0.0027777777777778 z)))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double tmp;
	if (t_0 <= -1e+282) {
		tmp = ((y / x) * z) * z;
	} else if (t_0 <= 5e+305) {
		tmp = fma((x - 0.5), log(x), ((fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = ((z / x) * z) * ((0.0007936500793651 + y) - (0.0027777777777778 / z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	tmp = 0.0
	if (t_0 <= -1e+282)
		tmp = Float64(Float64(Float64(y / x) * z) * z);
	elseif (t_0 <= 5e+305)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(Float64(0.0007936500793651 + y) - Float64(0.0027777777777778 / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+282], N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(N[(0.0007936500793651 + y), $MachinePrecision] - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000003e282

   1. Initial program 92.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 94.7%

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

   if -1.00000000000000003e282 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 5.00000000000000009e305

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.4

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.5

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.5

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 95.6

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

   9. Applied rewrites 95.6%

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

   if 5.00000000000000009e305 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 82.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.5%

      \[\leadsto \frac{z}{x} \cdot \color{blue}{-0.0027777777777778} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -1 \cdot 10^{+282}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(\left(0.0007936500793651 + y\right) - \frac{0.0027777777777778}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (/
        (+
         (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z)
         0.083333333333333)
        x)
       (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))
      2e+307)
   (fma
    (- x 0.5)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (fma (- x 0.5) (log x) (* (* (+ (/ 0.0007936500793651 x) (/ y x)) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467)) <= 2e+307) {
		tmp = fma((x - 0.5), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = fma((x - 0.5), log(x), ((((0.0007936500793651 / x) + (y / x)) * z) * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)) <= 2e+307)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * z) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 1.99999999999999997e307

   1. Initial program 97.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 97.9

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

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 98.0

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 98.0

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 98.0%

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

   if 1.99999999999999997e307 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 82.6

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

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 82.6

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 82.6

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 92.2

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

   9. Applied rewrites 92.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{0.5 + x}{\mathsf{fma}\left(x, x, -0.25\right)}}, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5000000000000.0)
   (fma
    (/ 1.0 (/ (+ 0.5 x) (fma x x -0.25)))
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (+
    (* (* (+ (/ 0.0007936500793651 x) (/ y x)) z) z)
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5000000000000.0) {
		tmp = fma((1.0 / ((0.5 + x) / fma(x, x, -0.25))), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = ((((0.0007936500793651 / x) + (y / x)) * z) * z) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5000000000000.0)
		tmp = fma(Float64(1.0 / Float64(Float64(0.5 + x) / fma(x, x, -0.25))), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * z) * z) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5000000000000.0], N[(N[(1.0 / N[(N[(0.5 + x), $MachinePrecision] / N[(x * x + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e12

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.7

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\frac{1}{2} + x}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)}}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}\right) \]

      11. metadata-eval 99.7

          \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{0.5 + x}{\mathsf{fma}\left(x, x, \color{blue}{-0.25}\right)}}, \log x, \left(0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right) \]

   8. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{0.5 + x}{\mathsf{fma}\left(x, x, -0.25\right)}}}, \log x, \left(0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right) \]

   if 5e12 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{0.5 + x}{\mathsf{fma}\left(x, x, -0.25\right)}}, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - \frac{0.5}{x}\right) \cdot x, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.6e+16)
   (fma
    (* (- 1.0 (/ 0.5 x)) x)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (+
    (* (* (+ (/ 0.0007936500793651 x) (/ y x)) z) z)
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.6e+16) {
		tmp = fma(((1.0 - (0.5 / x)) * x), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = ((((0.0007936500793651 / x) + (y / x)) * z) * z) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.6e+16)
		tmp = fma(Float64(Float64(1.0 - Float64(0.5 / x)) * x), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * z) * z) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.6e+16], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e16

   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
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.7

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.6e16 < x

   1. Initial program 86.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - \frac{0.5}{x}\right) \cdot x, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8000000000000:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8000000000000.0)
   (fma
    (- x 0.5)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (+
    (* (* (+ (/ 0.0007936500793651 x) (/ y x)) z) z)
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8000000000000.0) {
		tmp = fma((x - 0.5), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = ((((0.0007936500793651 / x) + (y / x)) * z) * z) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8000000000000.0)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * z) * z) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8000000000000.0], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e12

   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
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.7

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   if 8e12 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8000000000000:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.8e+175)
   (fma
    (- x 0.5)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (fma
    (- x 0.5)
    (log x)
    (-
     (fma
      (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
      z
      (+ (/ 0.083333333333333 x) 0.91893853320467))
     x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.8e+175) {
		tmp = fma((x - 0.5), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = fma((x - 0.5), log(x), (fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, ((0.083333333333333 / x) + 0.91893853320467)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.8e+175)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(Float64(0.083333333333333 / x) + 0.91893853320467)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.8e+175], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e175

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 97.8

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

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 97.8

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 97.8

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 97.9%

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

   if 5.8e175 < x

   1. Initial program 78.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 78.4

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

   4. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 78.5

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 78.5

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 78.5%

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

   7. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 92.4

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

   9. Applied rewrites 92.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right) + \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.8e+175)
   (fma
    (- x 0.5)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (-
    (+
     (fma
      (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
      z
      0.91893853320467)
     (fma (- x 0.5) (log x) (/ 0.083333333333333 x)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.8e+175) {
		tmp = fma((x - 0.5), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = (fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, 0.91893853320467) + fma((x - 0.5), log(x), (0.083333333333333 / x))) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.8e+175)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, 0.91893853320467) + fma(Float64(x - 0.5), log(x), Float64(0.083333333333333 / x))) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.8e+175], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + 0.91893853320467), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e175

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 97.8

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

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 97.8

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 97.8

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 97.9%

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

   if 5.8e175 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 92.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right) + \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 60000:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\left(z \cdot z\right) \cdot y}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 60000.0)
   (fma
    (- x 0.5)
    (log x)
    (/
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333)
     x))
   (if (<= x 8e+241)
     (fma (- x 0.5) (log x) (+ (/ (* (* z z) y) x) (- 0.91893853320467 x)))
     (fma
      (- x 0.5)
      (log x)
      (- (+ (/ 0.083333333333333 x) 0.91893853320467) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 60000.0) {
		tmp = fma((x - 0.5), log(x), (fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x));
	} else if (x <= 8e+241) {
		tmp = fma((x - 0.5), log(x), ((((z * z) * y) / x) + (0.91893853320467 - x)));
	} else {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 60000.0)
		tmp = fma(Float64(x - 0.5), log(x), Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x));
	elseif (x <= 8e+241)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(Float64(z * z) * y) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 60000.0], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+241], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6e4

   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
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.7

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 98.6

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

   9. Applied rewrites 98.6%

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

   if 6e4 < x < 8.0000000000000004e241

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.9

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 93.9

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) + \frac{\color{blue}{y \cdot {z}^{2}}}{x}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) + \frac{\color{blue}{{z}^{2} \cdot y}}{x}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) + \frac{\color{blue}{{z}^{2} \cdot y}}{x}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) + \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x}\right) \]

      4. lower-*.f64 87.8

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

   9. Applied rewrites 87.8%

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

   if 8.0000000000000004e241 < x

   1. Initial program 67.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x - \frac{1}{2}}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      7. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \color{blue}{\log x}, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      12. metadata-eval N/A

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

      13. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 60000:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+241}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\left(z \cdot z\right) \cdot y}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+268}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.5e+268)
   (fma
    (- x 0.5)
    (log x)
    (+
     (/
      (fma
       (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (- 0.91893853320467 x)))
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.5e+268) {
		tmp = fma((x - 0.5), log(x), ((fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.5e+268)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.5e+268], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.49999999999999972e268

   1. Initial program 96.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 96.1

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

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 96.1

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 96.1

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 96.2%

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

   if 3.49999999999999972e268 < x

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} - 1\right) \cdot x \]

      3. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) - 1\right) \cdot x \]

      4. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\log x - 1\right) \cdot x} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log x - 1\right)} \cdot x \]

      7. lower-log.f64 90.0

         \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x \]

   5. Applied rewrites 90.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+268}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.4e+45)
   (fma
    (- x 0.5)
    (log x)
    (/
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333)
     x))
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.4e+45) {
		tmp = fma((x - 0.5), log(x), (fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.4e+45)
		tmp = fma(Float64(x - 0.5), log(x), Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x));
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.4e+45], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.39999999999999968e45

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 99.1

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 99.1

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

      14. lift-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. unsub-neg N/A

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

      18. lower--.f64 99.1

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 95.2

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

   9. Applied rewrites 95.2%

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

   if 5.39999999999999968e45 < x

   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. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} - 1\right) \cdot x \]

      3. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) - 1\right) \cdot x \]

      4. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\log x - 1\right) \cdot x} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log x - 1\right)} \cdot x \]

      7. lower-log.f64 75.9

         \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x \]

   5. Applied rewrites 75.9%

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

Alternative 12: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.8e+45)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.8e+45) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.8e+45)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.8e+45], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8000000000000002e45

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if 3.8000000000000002e45 < x

   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. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} - 1\right) \cdot x \]

      3. log-rec N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) - 1\right) \cdot x \]

      4. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\log x - 1\right) \cdot x} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log x - 1\right)} \cdot x \]

      7. lower-log.f64 75.9

         \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x \]

   5. Applied rewrites 75.9%

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

Alternative 13: 65.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4e+36)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (+ (/ 0.0007936500793651 x) (/ y x)) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e+36) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((0.0007936500793651 / x) + (y / x)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4e+36)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4e+36], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e36

   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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 96.6

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0007936500793651 + y}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 1.4e36 < x

   1. Initial program 86.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 27.9

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (fma
   (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
   z
   0.083333333333333)
  x))

C

double code(double x, double y, double z) {
	return fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   9. lower-+.f64 63.5

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0007936500793651 + y}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

5. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]
6. Add Preprocessing

Alternative 15: 41.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right) \cdot z}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (* (fma (+ 0.0007936500793651 y) z -0.0027777777777778) z) x))

C

double code(double x, double y, double z) {
	return (fma((0.0007936500793651 + y), z, -0.0027777777777778) * z) / x;
}

Julia

function code(x, y, z)
	return Float64(Float64(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778) * z) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

6. Taylor expanded in z around inf

   \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{x} \]
7. Step-by-step derivation

8. Applied rewrites 45.8%

   \[\leadsto \frac{\left(\left(0.0007936500793651 + y\right) - \frac{0.0027777777777778}{z}\right) \cdot \left(z \cdot z\right)}{x} \]

9. Taylor expanded in z around 0

   \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]
10. Step-by-step derivation

11. Applied rewrites 46.3%

    \[\leadsto \frac{\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right) \cdot z}{x} \]
12. Add Preprocessing

Alternative 16: 41.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot z}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (* (+ 0.0007936500793651 y) z) z) x))

C

double code(double x, double y, double z) {
	return (((0.0007936500793651 + y) * z) * z) / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((0.0007936500793651d0 + y) * z) * z) / x
end function

Java

public static double code(double x, double y, double z) {
	return (((0.0007936500793651 + y) * z) * z) / x;
}

Python

def code(x, y, z):
	return (((0.0007936500793651 + y) * z) * z) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) * z) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((0.0007936500793651 + y) * z) * z) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

6. Taylor expanded in z around inf

   \[\leadsto \frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \]
7. Step-by-step derivation

8. Applied rewrites 46.0%

   \[\leadsto \frac{\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot z}{x} \]
9. Add Preprocessing

Alternative 17: 41.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (+ 0.0007936500793651 y) (* z z)) x))

C

double code(double x, double y, double z) {
	return ((0.0007936500793651 + y) * (z * z)) / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((0.0007936500793651d0 + y) * (z * z)) / x
end function

Java

public static double code(double x, double y, double z) {
	return ((0.0007936500793651 + y) * (z * z)) / x;
}

Python

def code(x, y, z):
	return ((0.0007936500793651 + y) * (z * z)) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z * z)) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((0.0007936500793651 + y) * (z * z)) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

6. Taylor expanded in z around inf

   \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{x} \]
7. Step-by-step derivation

8. Applied rewrites 45.8%

   \[\leadsto \frac{\left(\left(0.0007936500793651 + y\right) - \frac{0.0027777777777778}{z}\right) \cdot \left(z \cdot z\right)}{x} \]

9. Taylor expanded in z around inf

   \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \left(z \cdot z\right)}{x} \]
10. Step-by-step derivation

11. Applied rewrites 45.6%

    \[\leadsto \frac{\left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}{x} \]
12. Add Preprocessing

Alternative 18: 31.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{z \cdot z}{x} \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ (* z z) x) y))

C

double code(double x, double y, double z) {
	return ((z * z) / x) * y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((z * z) / x) * y
end function

Java

public static double code(double x, double y, double z) {
	return ((z * z) / x) * y;
}

Python

def code(x, y, z):
	return ((z * z) / x) * y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z * z) / x) * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z * z) / x) * y;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x} \]

   5. lower-*.f64 35.1

      \[\leadsto \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x} \]

5. Applied rewrites 35.1%

   \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot y}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 36.2%

   \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]

8. Final simplification 36.2%

   \[\leadsto \frac{z \cdot z}{x} \cdot y \]
9. Add Preprocessing

Alternative 19: 32.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{z}{x} \cdot z\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ z x) z) y))

C

double code(double x, double y, double z) {
	return ((z / x) * z) * y;
}

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) * z) * y
end function

Java

public static double code(double x, double y, double z) {
	return ((z / x) * z) * y;
}

Python

def code(x, y, z):
	return ((z / x) * z) * y

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * z) * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z / x) * z) * y;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x} \]

   5. lower-*.f64 35.1

      \[\leadsto \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x} \]

5. Applied rewrites 35.1%

   \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot y}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 36.2%

   \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
8. Step-by-step derivation

9. Applied rewrites 36.1%

   \[\leadsto y \cdot \left(\frac{z}{x} \cdot \color{blue}{z}\right) \]

10. Final simplification 36.1%

    \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot y \]
11. Add Preprocessing

Alternative 20: 8.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{z}{x} \cdot -0.0027777777777778 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ z x) -0.0027777777777778))

C

double code(double x, double y, double z) {
	return (z / x) * -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 = (z / x) * (-0.0027777777777778d0)
end function

Java

public static double code(double x, double y, double z) {
	return (z / x) * -0.0027777777777778;
}

Python

def code(x, y, z):
	return (z / x) * -0.0027777777777778

Julia

function code(x, y, z)
	return Float64(Float64(z / x) * -0.0027777777777778)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (z / x) * -0.0027777777777778;
end

Wolfram

code[x_, y_, z_] := N[(N[(z / x), $MachinePrecision] * -0.0027777777777778), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \cdot {z}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \cdot {z}^{2}} \]

   3. associate-*r/ N/A

      \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x \cdot z}}\right) \cdot {z}^{2} \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right) \cdot {z}^{2} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)} \cdot {z}^{2} \]

   6. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   9. associate-*r/ N/A

      \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{\color{blue}{z \cdot x}}\right) \cdot {z}^{2} \]

   13. associate-/r* N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \color{blue}{\frac{\frac{\frac{13888888888889}{5000000000000000}}{z}}{x}}\right) \cdot {z}^{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\color{blue}{\frac{13888888888889}{5000000000000000} \cdot 1}}{z}}{x}\right) \cdot {z}^{2} \]

   15. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}}}{x}\right) \cdot {z}^{2} \]

   16. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}}{x}}\right) \cdot {z}^{2} \]

   17. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{z}}}{x}\right) \cdot {z}^{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{z}}{x}\right) \cdot {z}^{2} \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{\frac{13888888888889}{5000000000000000}}{z}}}{x}\right) \cdot {z}^{2} \]

   20. unpow2 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\frac{13888888888889}{5000000000000000}}{z}}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   21. lower-*.f64 45.3

       \[\leadsto \left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) - \frac{\frac{0.0027777777777778}{z}}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

5. Applied rewrites 45.3%

   \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) - \frac{\frac{0.0027777777777778}{z}}{x}\right) \cdot \left(z \cdot z\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 9.9%

   \[\leadsto \frac{z}{x} \cdot \color{blue}{-0.0027777777777778} \]
9. Add Preprocessing

Alternative 21: 8.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.0027777777777778}{x} \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ -0.0027777777777778 x) z))

C

double code(double x, double y, double z) {
	return (-0.0027777777777778 / x) * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((-0.0027777777777778d0) / x) * z
end function

Java

public static double code(double x, double y, double z) {
	return (-0.0027777777777778 / x) * z;
}

Python

def code(x, y, z):
	return (-0.0027777777777778 / x) * z

Julia

function code(x, y, z)
	return Float64(Float64(-0.0027777777777778 / x) * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (-0.0027777777777778 / x) * z;
end

Wolfram

code[x_, y_, z_] := N[(N[(-0.0027777777777778 / x), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \cdot {z}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x \cdot z}\right) \cdot {z}^{2}} \]

   3. associate-*r/ N/A

      \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x \cdot z}}\right) \cdot {z}^{2} \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x \cdot z}\right) \cdot {z}^{2} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right)} \cdot {z}^{2} \]

   6. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   9. associate-*r/ N/A

      \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}}\right) - \frac{\frac{13888888888889}{5000000000000000}}{x \cdot z}\right) \cdot {z}^{2} \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{13888888888889}{5000000000000000}}{\color{blue}{z \cdot x}}\right) \cdot {z}^{2} \]

   13. associate-/r* N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \color{blue}{\frac{\frac{\frac{13888888888889}{5000000000000000}}{z}}{x}}\right) \cdot {z}^{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\color{blue}{\frac{13888888888889}{5000000000000000} \cdot 1}}{z}}{x}\right) \cdot {z}^{2} \]

   15. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}}}{x}\right) \cdot {z}^{2} \]

   16. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}}{x}}\right) \cdot {z}^{2} \]

   17. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{z}}}{x}\right) \cdot {z}^{2} \]

   18. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{z}}{x}\right) \cdot {z}^{2} \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\color{blue}{\frac{\frac{13888888888889}{5000000000000000}}{z}}}{x}\right) \cdot {z}^{2} \]

   20. unpow2 N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\frac{7936500793651}{10000000000000000}}{x}\right) - \frac{\frac{\frac{13888888888889}{5000000000000000}}{z}}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   21. lower-*.f64 45.3

       \[\leadsto \left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) - \frac{\frac{0.0027777777777778}{z}}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

5. Applied rewrites 45.3%

   \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) - \frac{\frac{0.0027777777777778}{z}}{x}\right) \cdot \left(z \cdot z\right)} \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{-13888888888889}{5000000000000000} \cdot \color{blue}{\frac{z}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 9.9%

   \[\leadsto \frac{z}{x} \cdot \color{blue}{-0.0027777777777778} \]
9. Step-by-step derivation

10. Applied rewrites 9.6%

    \[\leadsto z \cdot \frac{-0.0027777777777778}{\color{blue}{x}} \]

11. Final simplification 9.6%

    \[\leadsto \frac{-0.0027777777777778}{x} \cdot z \]
12. Add Preprocessing

Developer Target 1: 98.8% accurate, 0.9× 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 2024276 
(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)))