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

Percentage Accurate: 94.3% → 98.2%

Time: 19.6s

Alternatives: 17

Speedup: 1.0×

• 28.3% 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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+118}:\\ \;\;\;\;\left(\mathsf{fma}\left(x + -0.5, \log x, -x\right) + 0.91893853320467\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(z \cdot \frac{y}{x}\right) + \frac{z}{x} \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.9e+118)
   (+
    (+ (fma (+ x -0.5) (log x) (- x)) 0.91893853320467)
    (/
     (fma
      z
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      0.083333333333333)
     x))
   (+
    (- (* x (log x)) x)
    (+
     (/ 0.083333333333333 x)
     (+
      (* z (* z (/ y x)))
      (* (/ z x) (fma z 0.0007936500793651 -0.0027777777777778)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.9e+118) {
		tmp = (fma((x + -0.5), log(x), -x) + 0.91893853320467) + (fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = ((x * log(x)) - x) + ((0.083333333333333 / x) + ((z * (z * (y / x))) + ((z / x) * fma(z, 0.0007936500793651, -0.0027777777777778))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.9e+118)
		tmp = Float64(Float64(fma(Float64(x + -0.5), log(x), Float64(-x)) + 0.91893853320467) + Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(Float64(0.083333333333333 / x) + Float64(Float64(z * Float64(z * Float64(y / x))) + Float64(Float64(z / x) * fma(z, 0.0007936500793651, -0.0027777777777778)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.9e+118], N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.90000000000000016e118

   1. Initial program 99.6%

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

      1. fma-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if 2.90000000000000016e118 < x

   1. Initial program 85.6%

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

   2. Taylor expanded in x around inf 85.7%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 85.7%

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

      2. mul-1-neg 85.7%

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

      3. log-rec 85.7%

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

      4. remove-double-neg 85.7%

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

      5. metadata-eval 85.7%

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

      6. distribute-rgt-in 85.6%

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

      7. neg-mul-1 85.6%

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

      8. sub-neg 85.6%

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

      9. *-commutative 85.6%

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

   4. Simplified 85.6%

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

   5. Taylor expanded in y around 0 80.7%

      \[\leadsto \left(x \cdot \log x - x\right) + \color{blue}{\left(0.083333333333333 \cdot \frac{1}{x} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.7%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{x}} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      2. metadata-eval 80.7%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\frac{\color{blue}{0.083333333333333}}{x} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      3. *-commutative 80.7%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(\frac{\color{blue}{{z}^{2} \cdot y}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      4. associate-*r/ 85.9%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(\color{blue}{{z}^{2} \cdot \frac{y}{x}} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      5. unpow2 85.9%

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

      6. associate-*l* 88.6%

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

      7. associate-/l* 99.4%

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

      8. associate-/r/ 99.4%

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

      9. *-commutative 99.4%

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

      10. fma-neg 99.4%

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

      11. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+118}:\\ \;\;\;\;\left(\mathsf{fma}\left(x + -0.5, \log x, -x\right) + 0.91893853320467\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(z \cdot \frac{y}{x}\right) + \frac{z}{x} \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)\right)\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(z \cdot \frac{y}{x}\right) + \frac{z}{x} \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+81)
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x))
   (+
    (- (* x (log x)) x)
    (+
     (/ 0.083333333333333 x)
     (+
      (* z (* z (/ y x)))
      (* (/ z x) (fma z 0.0007936500793651 -0.0027777777777778)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+81) {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((x * log(x)) - x) + ((0.083333333333333 / x) + ((z * (z * (y / x))) + ((z / x) * fma(z, 0.0007936500793651, -0.0027777777777778))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+81)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(Float64(0.083333333333333 / x) + Float64(Float64(z * Float64(z * Float64(y / x))) + Float64(Float64(z / x) * fma(z, 0.0007936500793651, -0.0027777777777778)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+81], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999984e81

   1. Initial program 99.6%

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

   if 1.99999999999999984e81 < x

   1. Initial program 87.4%

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

   2. Taylor expanded in x around inf 87.5%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 87.5%

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

      2. mul-1-neg 87.5%

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

      3. log-rec 87.5%

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

      4. remove-double-neg 87.5%

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

      5. metadata-eval 87.5%

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

      6. distribute-rgt-in 87.4%

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

      7. neg-mul-1 87.4%

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

      8. sub-neg 87.4%

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

      9. *-commutative 87.4%

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

   4. Simplified 87.4%

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

   5. Taylor expanded in y around 0 83.2%

      \[\leadsto \left(x \cdot \log x - x\right) + \color{blue}{\left(0.083333333333333 \cdot \frac{1}{x} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 83.2%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{x}} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      2. metadata-eval 83.2%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\frac{\color{blue}{0.083333333333333}}{x} + \left(\frac{y \cdot {z}^{2}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      3. *-commutative 83.2%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(\frac{\color{blue}{{z}^{2} \cdot y}}{x} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      4. associate-*r/ 87.7%

         \[\leadsto \left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(\color{blue}{{z}^{2} \cdot \frac{y}{x}} + \frac{z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\right)\right) \]

      5. unpow2 87.7%

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

      6. associate-*l* 90.0%

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

      7. associate-/l* 99.4%

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

      8. associate-/r/ 99.4%

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

      9. *-commutative 99.4%

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

      10. fma-neg 99.4%

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

      11. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(z \cdot \frac{y}{x}\right) + \frac{z}{x} \cdot \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)\right)\right)\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+121}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.8e+121)
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x))
   (+ (- (* x (log x)) x) (* z (* z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.8e+121) {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((x * log(x)) - x) + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 5.8d+121) then
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x)
    else
        tmp = ((x * log(x)) - x) + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.8e+121) {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = ((x * Math.log(x)) - x) + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5.8e+121:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x)
	else:
		tmp = ((x * math.log(x)) - x) + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.8e+121)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5.8e+121)
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	else
		tmp = ((x * log(x)) - x) + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.8e+121], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.7999999999999998e121

   1. Initial program 99.6%

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

   if 5.7999999999999998e121 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf 85.4%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 85.4%

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

      2. mul-1-neg 85.4%

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

      3. log-rec 85.4%

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

      4. remove-double-neg 85.4%

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

      5. metadata-eval 85.4%

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

      6. distribute-rgt-in 85.3%

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

      7. neg-mul-1 85.3%

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

      8. sub-neg 85.3%

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

      9. *-commutative 85.3%

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

   4. Simplified 85.3%

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

   5. Taylor expanded in y around inf 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r/ 88.0%

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

      3. unpow2 88.0%

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

      4. associate-*l* 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+121}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 4: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+122}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+122)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x)
    (* x (+ (log x) -1.0)))
   (+ (- (* x (log x)) x) (* z (* z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+122) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (x * (log(x) + -1.0));
	} else {
		tmp = ((x * log(x)) - x) + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1d+122) then
        tmp = ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x) + (x * (log(x) + (-1.0d0)))
    else
        tmp = ((x * log(x)) - x) + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+122) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (x * (Math.log(x) + -1.0));
	} else {
		tmp = ((x * Math.log(x)) - x) + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1e+122:
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (x * (math.log(x) + -1.0))
	else:
		tmp = ((x * math.log(x)) - x) + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+122)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x) + Float64(x * Float64(log(x) + -1.0)));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1e+122)
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (x * (log(x) + -1.0));
	else
		tmp = ((x * log(x)) - x) + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+122], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000001e122

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 47.6%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      2. mul-1-neg 47.6%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      3. log-rec 47.6%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      4. remove-double-neg 47.6%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      5. metadata-eval 47.6%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   4. Simplified 99.1%

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

   if 1.00000000000000001e122 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf 85.4%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 85.4%

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

      2. mul-1-neg 85.4%

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

      3. log-rec 85.4%

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

      4. remove-double-neg 85.4%

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

      5. metadata-eval 85.4%

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

      6. distribute-rgt-in 85.3%

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

      7. neg-mul-1 85.3%

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

      8. sub-neg 85.3%

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

      9. *-commutative 85.3%

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

   4. Simplified 85.3%

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

   5. Taylor expanded in y around inf 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r/ 88.0%

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

      3. unpow2 88.0%

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

      4. associate-*l* 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+122}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 5: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.58 \cdot 10^{-65} \lor \neg \left(z \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.58e-65) (not (<= z 2e-11)))
   (+ (- (* x (log x)) x) (/ (* z z) (/ x (+ y 0.0007936500793651))))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* 0.083333333333333 (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.58e-65) || !(z <= 2e-11)) {
		tmp = ((x * log(x)) - x) + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.58d-65)) .or. (.not. (z <= 2d-11))) then
        tmp = ((x * log(x)) - x) + ((z * z) / (x / (y + 0.0007936500793651d0)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (0.083333333333333d0 * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.58e-65) || !(z <= 2e-11)) {
		tmp = ((x * Math.log(x)) - x) + ((z * z) / (x / (y + 0.0007936500793651)));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.58e-65) or not (z <= 2e-11):
		tmp = ((x * math.log(x)) - x) + ((z * z) / (x / (y + 0.0007936500793651)))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.58e-65) || !(z <= 2e-11))
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(0.083333333333333 * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.58e-65) || ~((z <= 2e-11)))
		tmp = ((x * log(x)) - x) + ((z * z) / (x / (y + 0.0007936500793651)));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.58e-65], N[Not[LessEqual[z, 2e-11]], $MachinePrecision]], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.58000000000000005e-65 or 1.99999999999999988e-11 < z

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 91.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 91.9%

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

      2. mul-1-neg 91.9%

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

      3. log-rec 91.9%

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

      4. remove-double-neg 91.9%

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

      5. metadata-eval 91.9%

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

      6. distribute-rgt-in 91.9%

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

      7. neg-mul-1 91.9%

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

      8. sub-neg 91.9%

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

      9. *-commutative 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in z around inf 90.2%

      \[\leadsto \left(x \cdot \log x - x\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto \left(x \cdot \log x - x\right) + \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}} \]

      2. unpow2 93.3%

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

   7. Simplified 93.3%

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

   if -1.58000000000000005e-65 < z < 1.99999999999999988e-11

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 98.2%

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

      1. div-inv 51.3%

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

   4. Applied egg-rr 98.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{0.083333333333333 \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.58 \cdot 10^{-65} \lor \neg \left(z \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log x - x\\ \mathbf{if}\;z \leq -6 \cdot 10^{-64}:\\ \;\;\;\;t_0 + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log x)) x)))
   (if (<= z -6e-64)
     (+ t_0 (/ y (/ x (* z z))))
     (if (<= z 2.8e-10)
       (+
        (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
        (* 0.083333333333333 (/ 1.0 x)))
       (+ t_0 (* z (* z (/ y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log(x)) - x;
	double tmp;
	if (z <= -6e-64) {
		tmp = t_0 + (y / (x / (z * z)));
	} else if (z <= 2.8e-10) {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	} else {
		tmp = t_0 + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * log(x)) - x
    if (z <= (-6d-64)) then
        tmp = t_0 + (y / (x / (z * z)))
    else if (z <= 2.8d-10) then
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (0.083333333333333d0 * (1.0d0 / x))
    else
        tmp = t_0 + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log(x)) - x;
	double tmp;
	if (z <= -6e-64) {
		tmp = t_0 + (y / (x / (z * z)));
	} else if (z <= 2.8e-10) {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	} else {
		tmp = t_0 + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log(x)) - x
	tmp = 0
	if z <= -6e-64:
		tmp = t_0 + (y / (x / (z * z)))
	elif z <= 2.8e-10:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x))
	else:
		tmp = t_0 + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(x)) - x)
	tmp = 0.0
	if (z <= -6e-64)
		tmp = Float64(t_0 + Float64(y / Float64(x / Float64(z * z))));
	elseif (z <= 2.8e-10)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(0.083333333333333 * Float64(1.0 / x)));
	else
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log(x)) - x;
	tmp = 0.0;
	if (z <= -6e-64)
		tmp = t_0 + (y / (x / (z * z)));
	elseif (z <= 2.8e-10)
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	else
		tmp = t_0 + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -6e-64], N[(t$95$0 + N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e-10], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.0000000000000001e-64

   1. Initial program 96.5%

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

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. mul-1-neg 96.6%

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

      3. log-rec 96.6%

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

      4. remove-double-neg 96.6%

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

      5. metadata-eval 96.6%

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

      6. distribute-rgt-in 96.5%

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

      7. neg-mul-1 96.5%

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

      8. sub-neg 96.5%

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

      9. *-commutative 96.5%

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

   4. Simplified 96.5%

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

   5. Taylor expanded in y around inf 70.5%

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

      1. associate-/l* 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   if -6.0000000000000001e-64 < z < 2.80000000000000015e-10

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 98.2%

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

      1. div-inv 51.3%

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

   4. Applied egg-rr 98.3%

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

   if 2.80000000000000015e-10 < z

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 86.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 86.0%

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

      2. mul-1-neg 86.0%

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

      3. log-rec 86.0%

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

      4. remove-double-neg 86.0%

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

      5. metadata-eval 86.0%

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

      6. distribute-rgt-in 85.9%

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

      7. neg-mul-1 85.9%

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

      8. sub-neg 85.9%

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

      9. *-commutative 85.9%

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

   4. Simplified 85.9%

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

   5. Taylor expanded in y around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*r/ 75.9%

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

      3. unpow2 75.9%

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

      4. associate-*l* 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 7: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log x - x\\ \mathbf{if}\;z \leq -6 \cdot 10^{-64}:\\ \;\;\;\;t_0 + \frac{1}{\frac{\frac{x}{z \cdot z}}{y + 0.0007936500793651}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log x)) x)))
   (if (<= z -6e-64)
     (+ t_0 (/ 1.0 (/ (/ x (* z z)) (+ y 0.0007936500793651))))
     (if (<= z 1.8e-10)
       (+
        (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
        (* 0.083333333333333 (/ 1.0 x)))
       (+ t_0 (/ (* z z) (/ x (+ y 0.0007936500793651))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log(x)) - x;
	double tmp;
	if (z <= -6e-64) {
		tmp = t_0 + (1.0 / ((x / (z * z)) / (y + 0.0007936500793651)));
	} else if (z <= 1.8e-10) {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	} else {
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * log(x)) - x
    if (z <= (-6d-64)) then
        tmp = t_0 + (1.0d0 / ((x / (z * z)) / (y + 0.0007936500793651d0)))
    else if (z <= 1.8d-10) then
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (0.083333333333333d0 * (1.0d0 / x))
    else
        tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log(x)) - x;
	double tmp;
	if (z <= -6e-64) {
		tmp = t_0 + (1.0 / ((x / (z * z)) / (y + 0.0007936500793651)));
	} else if (z <= 1.8e-10) {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	} else {
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log(x)) - x
	tmp = 0
	if z <= -6e-64:
		tmp = t_0 + (1.0 / ((x / (z * z)) / (y + 0.0007936500793651)))
	elif z <= 1.8e-10:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x))
	else:
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(x)) - x)
	tmp = 0.0
	if (z <= -6e-64)
		tmp = Float64(t_0 + Float64(1.0 / Float64(Float64(x / Float64(z * z)) / Float64(y + 0.0007936500793651))));
	elseif (z <= 1.8e-10)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(0.083333333333333 * Float64(1.0 / x)));
	else
		tmp = Float64(t_0 + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log(x)) - x;
	tmp = 0.0;
	if (z <= -6e-64)
		tmp = t_0 + (1.0 / ((x / (z * z)) / (y + 0.0007936500793651)));
	elseif (z <= 1.8e-10)
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 * (1.0 / x));
	else
		tmp = t_0 + ((z * z) / (x / (y + 0.0007936500793651)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -6e-64], N[(t$95$0 + N[(1.0 / N[(N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e-10], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.0000000000000001e-64

   1. Initial program 96.5%

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

      1. clear-num 96.5%

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

      2. inv-pow 96.5%

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

      3. *-commutative 96.5%

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

      4. fma-udef 96.5%

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

      5. fma-neg 96.5%

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

      6. metadata-eval 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in z around inf 93.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + {\color{blue}{\left(\frac{x}{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}\right)}}^{-1} \]
   5. Step-by-step derivation

      1. associate-/r* 94.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + {\color{blue}{\left(\frac{\frac{x}{{z}^{2}}}{0.0007936500793651 + y}\right)}}^{-1} \]

      2. unpow2 94.8%

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

   6. Simplified 94.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + {\color{blue}{\left(\frac{\frac{x}{z \cdot z}}{0.0007936500793651 + y}\right)}}^{-1} \]
   7. Step-by-step derivation

      1. unpow-1 94.8%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in x around inf 94.9%

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

       1. sub-neg 96.6%

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

       2. mul-1-neg 96.6%

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

       3. log-rec 96.6%

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

       4. remove-double-neg 96.6%

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

       5. metadata-eval 96.6%

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

       6. distribute-rgt-in 96.5%

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

       7. neg-mul-1 96.5%

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

       8. sub-neg 96.5%

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

       9. *-commutative 96.5%

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

   11. Simplified 94.8%

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

   if -6.0000000000000001e-64 < z < 1.8e-10

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 98.2%

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

      1. div-inv 51.3%

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

   4. Applied egg-rr 98.3%

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

   if 1.8e-10 < z

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 86.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 86.0%

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

      2. mul-1-neg 86.0%

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

      3. log-rec 86.0%

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

      4. remove-double-neg 86.0%

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

      5. metadata-eval 86.0%

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

      6. distribute-rgt-in 85.9%

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

      7. neg-mul-1 85.9%

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

      8. sub-neg 85.9%

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

      9. *-commutative 85.9%

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

   4. Simplified 85.9%

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

   5. Taylor expanded in z around inf 85.8%

      \[\leadsto \left(x \cdot \log x - x\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 91.5%

         \[\leadsto \left(x \cdot \log x - x\right) + \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}} \]

      2. unpow2 91.5%

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

   7. Simplified 91.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{1}{\frac{\frac{x}{z \cdot z}}{y + 0.0007936500793651}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \end{array} \]

Alternative 8: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log x - x\\ \mathbf{if}\;x \leq 4.8 \cdot 10^{+123}:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log x)) x)))
   (if (<= x 4.8e+123)
     (+ t_0 (/ (+ 0.083333333333333 (* (+ y 0.0007936500793651) (* z z))) x))
     (+ t_0 (* z (* z (/ y x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log(x)) - x;
	double tmp;
	if (x <= 4.8e+123) {
		tmp = t_0 + ((0.083333333333333 + ((y + 0.0007936500793651) * (z * z))) / x);
	} else {
		tmp = t_0 + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * log(x)) - x
    if (x <= 4.8d+123) then
        tmp = t_0 + ((0.083333333333333d0 + ((y + 0.0007936500793651d0) * (z * z))) / x)
    else
        tmp = t_0 + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log(x)) - x;
	double tmp;
	if (x <= 4.8e+123) {
		tmp = t_0 + ((0.083333333333333 + ((y + 0.0007936500793651) * (z * z))) / x);
	} else {
		tmp = t_0 + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log(x)) - x
	tmp = 0
	if x <= 4.8e+123:
		tmp = t_0 + ((0.083333333333333 + ((y + 0.0007936500793651) * (z * z))) / x)
	else:
		tmp = t_0 + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(x)) - x)
	tmp = 0.0
	if (x <= 4.8e+123)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(Float64(y + 0.0007936500793651) * Float64(z * z))) / x));
	else
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log(x)) - x;
	tmp = 0.0;
	if (x <= 4.8e+123)
		tmp = t_0 + ((0.083333333333333 + ((y + 0.0007936500793651) * (z * z))) / x);
	else
		tmp = t_0 + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 4.8e+123], N[(t$95$0 + N[(N[(0.083333333333333 + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999978e123

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 99.1%

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

      2. mul-1-neg 99.1%

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

      3. log-rec 99.1%

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

      4. remove-double-neg 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-rgt-in 99.1%

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

      7. neg-mul-1 99.1%

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

      8. sub-neg 99.1%

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

      9. *-commutative 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in z around inf 98.0%

      \[\leadsto \left(x \cdot \log x - x\right) + \frac{\color{blue}{{z}^{2} \cdot \left(0.0007936500793651 + y\right)} + 0.083333333333333}{x} \]
   6. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \left(x \cdot \log x - x\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}} + 0.083333333333333}{x} \]

      2. unpow2 98.0%

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

   7. Simplified 98.0%

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

   if 4.79999999999999978e123 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in x around inf 85.4%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 85.4%

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

      2. mul-1-neg 85.4%

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

      3. log-rec 85.4%

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

      4. remove-double-neg 85.4%

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

      5. metadata-eval 85.4%

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

      6. distribute-rgt-in 85.3%

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

      7. neg-mul-1 85.3%

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

      8. sub-neg 85.3%

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

      9. *-commutative 85.3%

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

   4. Simplified 85.3%

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

   5. Taylor expanded in y around inf 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*r/ 88.0%

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

      3. unpow2 88.0%

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

      4. associate-*l* 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+123}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{0.083333333333333 + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 9: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-64} \lor \neg \left(z \leq 2.55 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-64) (not (<= z 2.55e-11)))
   (+ (- (* x (log x)) x) (* z (* z (/ y x))))
   (+
    (/ 0.083333333333333 x)
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-64) || !(z <= 2.55e-11)) {
		tmp = ((x * log(x)) - x) + (z * (z * (y / x)));
	} else {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1d-64)) .or. (.not. (z <= 2.55d-11))) then
        tmp = ((x * log(x)) - x) + (z * (z * (y / x)))
    else
        tmp = (0.083333333333333d0 / x) + (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-64) || !(z <= 2.55e-11)) {
		tmp = ((x * Math.log(x)) - x) + (z * (z * (y / x)));
	} else {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e-64) or not (z <= 2.55e-11):
		tmp = ((x * math.log(x)) - x) + (z * (z * (y / x)))
	else:
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-64) || !(z <= 2.55e-11))
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(z * Float64(z * Float64(y / x))));
	else
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1e-64) || ~((z <= 2.55e-11)))
		tmp = ((x * log(x)) - x) + (z * (z * (y / x)));
	else
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-64], N[Not[LessEqual[z, 2.55e-11]], $MachinePrecision]], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999965e-65 or 2.54999999999999992e-11 < z

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 91.9%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 91.9%

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

      2. mul-1-neg 91.9%

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

      3. log-rec 91.9%

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

      4. remove-double-neg 91.9%

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

      5. metadata-eval 91.9%

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

      6. distribute-rgt-in 91.9%

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

      7. neg-mul-1 91.9%

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

      8. sub-neg 91.9%

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

      9. *-commutative 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in y around inf 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*r/ 73.4%

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

      3. unpow2 73.4%

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

      4. associate-*l* 75.1%

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

   7. Simplified 75.1%

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

   if -9.99999999999999965e-65 < z < 2.54999999999999992e-11

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 98.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-64} \lor \neg \left(z \leq 2.55 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \]

Alternative 10: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log x - x\\ \mathbf{if}\;z \leq -6 \cdot 10^{-64}:\\ \;\;\;\;t_0 + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log x)) x)))
   (if (<= z -6e-64)
     (+ t_0 (/ y (/ x (* z z))))
     (if (<= z 1.12e-10)
       (+
        (/ 0.083333333333333 x)
        (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))
       (+ t_0 (* z (* z (/ y x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log(x)) - x;
	double tmp;
	if (z <= -6e-64) {
		tmp = t_0 + (y / (x / (z * z)));
	} else if (z <= 1.12e-10) {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	} else {
		tmp = t_0 + (z * (z * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * log(x)) - x
    if (z <= (-6d-64)) then
        tmp = t_0 + (y / (x / (z * z)))
    else if (z <= 1.12d-10) then
        tmp = (0.083333333333333d0 / x) + (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x))
    else
        tmp = t_0 + (z * (z * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log(x)) - x;
	double tmp;
	if (z <= -6e-64) {
		tmp = t_0 + (y / (x / (z * z)));
	} else if (z <= 1.12e-10) {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x));
	} else {
		tmp = t_0 + (z * (z * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log(x)) - x
	tmp = 0
	if z <= -6e-64:
		tmp = t_0 + (y / (x / (z * z)))
	elif z <= 1.12e-10:
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x))
	else:
		tmp = t_0 + (z * (z * (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(x)) - x)
	tmp = 0.0
	if (z <= -6e-64)
		tmp = Float64(t_0 + Float64(y / Float64(x / Float64(z * z))));
	elseif (z <= 1.12e-10)
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	else
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log(x)) - x;
	tmp = 0.0;
	if (z <= -6e-64)
		tmp = t_0 + (y / (x / (z * z)));
	elseif (z <= 1.12e-10)
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	else
		tmp = t_0 + (z * (z * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -6e-64], N[(t$95$0 + N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.12e-10], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.0000000000000001e-64

   1. Initial program 96.5%

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

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 96.6%

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

      2. mul-1-neg 96.6%

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

      3. log-rec 96.6%

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

      4. remove-double-neg 96.6%

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

      5. metadata-eval 96.6%

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

      6. distribute-rgt-in 96.5%

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

      7. neg-mul-1 96.5%

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

      8. sub-neg 96.5%

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

      9. *-commutative 96.5%

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

   4. Simplified 96.5%

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

   5. Taylor expanded in y around inf 70.5%

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

      1. associate-/l* 72.7%

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

      2. unpow2 72.7%

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

   7. Simplified 72.7%

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

   if -6.0000000000000001e-64 < z < 1.12e-10

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0 98.2%

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

   if 1.12e-10 < z

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 86.0%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 86.0%

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

      2. mul-1-neg 86.0%

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

      3. log-rec 86.0%

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

      4. remove-double-neg 86.0%

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

      5. metadata-eval 86.0%

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

      6. distribute-rgt-in 85.9%

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

      7. neg-mul-1 85.9%

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

      8. sub-neg 85.9%

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

      9. *-commutative 85.9%

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

   4. Simplified 85.9%

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

   5. Taylor expanded in y around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*r/ 75.9%

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

      3. unpow2 75.9%

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

      4. associate-*l* 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 11: 61.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-43}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + -0.0027777777777778 \cdot \frac{z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.55e-43)
   (+ (- (* x (log x)) x) (* -0.0027777777777778 (/ z x)))
   (+
    (/ 0.083333333333333 x)
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.55e-43) {
		tmp = ((x * log(x)) - x) + (-0.0027777777777778 * (z / x));
	} else {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.55d-43)) then
        tmp = ((x * log(x)) - x) + ((-0.0027777777777778d0) * (z / x))
    else
        tmp = (0.083333333333333d0 / x) + (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.55e-43) {
		tmp = ((x * Math.log(x)) - x) + (-0.0027777777777778 * (z / x));
	} else {
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.55e-43:
		tmp = ((x * math.log(x)) - x) + (-0.0027777777777778 * (z / x))
	else:
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.55e-43)
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(-0.0027777777777778 * Float64(z / x)));
	else
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.55e-43)
		tmp = ((x * log(x)) - x) + (-0.0027777777777778 * (z / x));
	else
		tmp = (0.083333333333333 / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.55e-43], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(-0.0027777777777778 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5499999999999998e-43

   1. Initial program 96.2%

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

   2. Taylor expanded in x around inf 96.3%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 96.3%

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

      2. mul-1-neg 96.3%

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

      3. log-rec 96.3%

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

      4. remove-double-neg 96.3%

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

      5. metadata-eval 96.3%

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

      6. distribute-rgt-in 96.2%

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

      7. neg-mul-1 96.2%

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

      8. sub-neg 96.2%

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

      9. *-commutative 96.2%

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

   4. Simplified 96.2%

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

   5. Taylor expanded in z around 0 51.2%

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

      1. *-commutative 51.2%

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

   7. Simplified 51.2%

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

   8. Taylor expanded in z around inf 52.2%

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

   if -2.5499999999999998e-43 < z

   1. Initial program 94.2%

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

   2. Taylor expanded in z around 0 69.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-43}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + -0.0027777777777778 \cdot \frac{z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \end{array} \]

Alternative 12: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\sqrt[3]{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+125)
   (cbrt
    (*
     (/ 0.083333333333333 x)
     (* (/ 0.083333333333333 x) (/ 0.083333333333333 x))))
   (+ (/ 0.083333333333333 x) (* x (+ (log x) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+125) {
		tmp = cbrt(((0.083333333333333 / x) * ((0.083333333333333 / x) * (0.083333333333333 / x))));
	} else {
		tmp = (0.083333333333333 / x) + (x * (log(x) + -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+125) {
		tmp = Math.cbrt(((0.083333333333333 / x) * ((0.083333333333333 / x) * (0.083333333333333 / x))));
	} else {
		tmp = (0.083333333333333 / x) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+125)
		tmp = cbrt(Float64(Float64(0.083333333333333 / x) * Float64(Float64(0.083333333333333 / x) * Float64(0.083333333333333 / x))));
	else
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+125], N[Power[N[(N[(0.083333333333333 / x), $MachinePrecision] * N[(N[(0.083333333333333 / x), $MachinePrecision] * N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1e125

   1. Initial program 95.1%

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

   2. Taylor expanded in z around 0 5.4%

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

   3. Taylor expanded in x around inf 5.4%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. sub-neg 5.4%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      2. mul-1-neg 5.4%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      3. log-rec 5.4%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      4. remove-double-neg 5.4%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      5. metadata-eval 5.4%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   5. Simplified 5.4%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

   6. Taylor expanded in x around 0 3.7%

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

      1. add-cbrt-cube 40.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right) \cdot \frac{0.083333333333333}{x}}} \]

   8. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right) \cdot \frac{0.083333333333333}{x}}} \]
   9. Step-by-step derivation

      1. associate-*l* 40.6%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}} \]

   10. Simplified 40.6%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}} \]

   if -3.1e125 < z

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0 67.0%

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

   3. Taylor expanded in x around inf 66.7%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. sub-neg 66.7%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      2. mul-1-neg 66.7%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      3. log-rec 66.7%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      4. remove-double-neg 66.7%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      5. metadata-eval 66.7%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+125}:\\ \;\;\;\;\sqrt[3]{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + x \cdot \left(\log x + -1\right)\\ \end{array} \]

Alternative 13: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+126}:\\ \;\;\;\;\sqrt[3]{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.12e+126)
   (cbrt
    (*
     (/ 0.083333333333333 x)
     (* (/ 0.083333333333333 x) (/ 0.083333333333333 x))))
   (+ (* x (+ (log x) -1.0)) (* 0.083333333333333 (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.12e+126) {
		tmp = cbrt(((0.083333333333333 / x) * ((0.083333333333333 / x) * (0.083333333333333 / x))));
	} else {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.12e+126) {
		tmp = Math.cbrt(((0.083333333333333 / x) * ((0.083333333333333 / x) * (0.083333333333333 / x))));
	} else {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.12e+126)
		tmp = cbrt(Float64(Float64(0.083333333333333 / x) * Float64(Float64(0.083333333333333 / x) * Float64(0.083333333333333 / x))));
	else
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.12e+126], N[Power[N[(N[(0.083333333333333 / x), $MachinePrecision] * N[(N[(0.083333333333333 / x), $MachinePrecision] * N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12e126

   1. Initial program 95.1%

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

   2. Taylor expanded in z around 0 5.4%

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

   3. Taylor expanded in x around inf 5.4%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. sub-neg 5.4%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      2. mul-1-neg 5.4%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      3. log-rec 5.4%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      4. remove-double-neg 5.4%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      5. metadata-eval 5.4%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   5. Simplified 5.4%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

   6. Taylor expanded in x around 0 3.7%

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

      1. add-cbrt-cube 40.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right) \cdot \frac{0.083333333333333}{x}}} \]

   8. Applied egg-rr 40.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right) \cdot \frac{0.083333333333333}{x}}} \]
   9. Step-by-step derivation

      1. associate-*l* 40.6%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}} \]

   10. Simplified 40.6%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}} \]

   if -1.12e126 < z

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0 67.0%

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

   3. Taylor expanded in x around inf 66.7%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. sub-neg 66.7%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      2. mul-1-neg 66.7%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      3. log-rec 66.7%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      4. remove-double-neg 66.7%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      5. metadata-eval 66.7%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

   6. Taylor expanded in x around 0 66.7%

      \[\leadsto \color{blue}{x \cdot \left(\log x - 1\right) + 0.083333333333333 \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+126}:\\ \;\;\;\;\sqrt[3]{\frac{0.083333333333333}{x} \cdot \left(\frac{0.083333333333333}{x} \cdot \frac{0.083333333333333}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 14: 60.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -38:\\ \;\;\;\;\left(x \cdot \log x - x\right) + -0.0027777777777778 \cdot \frac{z}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -38.0)
   (+ (- (* x (log x)) x) (* -0.0027777777777778 (/ z x)))
   (+ (* x (+ (log x) -1.0)) (* 0.083333333333333 (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -38.0) {
		tmp = ((x * log(x)) - x) + (-0.0027777777777778 * (z / x));
	} else {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-38.0d0)) then
        tmp = ((x * log(x)) - x) + ((-0.0027777777777778d0) * (z / x))
    else
        tmp = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -38.0) {
		tmp = ((x * Math.log(x)) - x) + (-0.0027777777777778 * (z / x));
	} else {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 * (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -38.0:
		tmp = ((x * math.log(x)) - x) + (-0.0027777777777778 * (z / x))
	else:
		tmp = (x * (math.log(x) + -1.0)) + (0.083333333333333 * (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -38.0)
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(-0.0027777777777778 * Float64(z / x)));
	else
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -38.0)
		tmp = ((x * log(x)) - x) + (-0.0027777777777778 * (z / x));
	else
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -38.0], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(-0.0027777777777778 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -38

   1. Initial program 95.7%

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

   2. Taylor expanded in x around inf 95.8%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. sub-neg 95.8%

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

      2. mul-1-neg 95.8%

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

      3. log-rec 95.8%

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

      4. remove-double-neg 95.8%

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

      5. metadata-eval 95.8%

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

      6. distribute-rgt-in 95.7%

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

      7. neg-mul-1 95.7%

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

      8. sub-neg 95.7%

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

      9. *-commutative 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in z around 0 47.6%

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

      1. *-commutative 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in z around inf 48.8%

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

   if -38 < z

   1. Initial program 94.5%

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

   2. Taylor expanded in z around 0 69.8%

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

   3. Taylor expanded in x around inf 69.4%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. sub-neg 69.4%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      2. mul-1-neg 69.4%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      3. log-rec 69.4%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      4. remove-double-neg 69.4%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

      5. metadata-eval 69.4%

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

   6. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{x \cdot \left(\log x - 1\right) + 0.083333333333333 \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -38:\\ \;\;\;\;\left(x \cdot \log x - x\right) + -0.0027777777777778 \cdot \frac{z}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 15: 56.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} + x \cdot \left(\log x + -1\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (/ 0.083333333333333 x) (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	return (0.083333333333333 / x) + (x * (log(x) + -1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (0.083333333333333d0 / x) + (x * (log(x) + (-1.0d0)))
end function

Java

public static double code(double x, double y, double z) {
	return (0.083333333333333 / x) + (x * (Math.log(x) + -1.0));
}

Python

def code(x, y, z):
	return (0.083333333333333 / x) + (x * (math.log(x) + -1.0))

Julia

function code(x, y, z)
	return Float64(Float64(0.083333333333333 / x) + Float64(x * Float64(log(x) + -1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.083333333333333 / x) + (x * (log(x) + -1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

2. Taylor expanded in z around 0 57.9%

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

3. Taylor expanded in x around inf 57.6%

   \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
4. Step-by-step derivation

   1. sub-neg 57.6%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

   2. mul-1-neg 57.6%

      \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   3. log-rec 57.6%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   4. remove-double-neg 57.6%

      \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   5. metadata-eval 57.6%

      \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

6. Final simplification 57.6%

   \[\leadsto \frac{0.083333333333333}{x} + x \cdot \left(\log x + -1\right) \]

Alternative 16: 23.4% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ 0.083333333333333 \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.083333333333333 (/ 1.0 x)))

C

double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 * (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Python

def code(x, y, z):
	return 0.083333333333333 * (1.0 / x)

Julia

function code(x, y, z)
	return Float64(0.083333333333333 * Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 * (1.0 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

2. Taylor expanded in z around 0 57.9%

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

3. Taylor expanded in x around inf 57.6%

   \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
4. Step-by-step derivation

   1. sub-neg 57.6%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

   2. mul-1-neg 57.6%

      \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   3. log-rec 57.6%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   4. remove-double-neg 57.6%

      \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   5. metadata-eval 57.6%

      \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

6. Taylor expanded in x around 0 23.1%

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

   1. div-inv 23.2%

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

8. Applied egg-rr 23.2%

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

9. Final simplification 23.2%

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

Alternative 17: 23.4% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

2. Taylor expanded in z around 0 57.9%

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

3. Taylor expanded in x around inf 57.6%

   \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]
4. Step-by-step derivation

   1. sub-neg 57.6%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

   2. mul-1-neg 57.6%

      \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   3. log-rec 57.6%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   4. remove-double-neg 57.6%

      \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(-1\right)\right) + \frac{0.083333333333333}{x} \]

   5. metadata-eval 57.6%

      \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

6. Taylor expanded in x around 0 23.1%

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

7. Final simplification 23.1%

   \[\leadsto \frac{0.083333333333333}{x} \]

Developer target: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))