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

Percentage Accurate: 93.9% → 98.8%

Time: 13.4s

Alternatives: 14

Speedup: 1.0×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-110}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) - \frac{0.0027777777777778}{x}, \log x \cdot \left(x + -0.5\right)\right)\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e-110)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     1.0
     (/
      x
      (fma
       z
       (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
       0.083333333333333))))
   (-
    (+
     0.91893853320467
     (+
      (/ 0.083333333333333 x)
      (fma
       z
       (- (* z (+ (/ 0.0007936500793651 x) (/ y x))) (/ 0.0027777777777778 x))
       (* (log x) (+ x -0.5)))))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-110) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + (1.0 / (x / fma(z, fma(z, (0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333)));
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + fma(z, ((z * ((0.0007936500793651 / x) + (y / x))) - (0.0027777777777778 / x)), (log(x) * (x + -0.5))))) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e-110)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(1.0 / Float64(x / fma(z, fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), 0.083333333333333))));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 / x) + fma(z, Float64(Float64(z * Float64(Float64(0.0007936500793651 / x) + Float64(y / x))) - Float64(0.0027777777777778 / x)), Float64(log(x) * Float64(x + -0.5))))) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e-110], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(1.0 / N[(x / N[(z * N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(z * N[(N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e-110

   1. Initial program 99.7%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. *-commutative 99.8%

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

      4. fma-undefine 99.8%

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

      5. fmm-def 99.8%

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

      6. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. fma-define 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. fma-define 99.8%

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

   6. Simplified 99.8%

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

   if 1.0000000000000001e-110 < x

   1. Initial program 91.4%

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

      1. remove-double-neg 91.4%

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

      2. distribute-frac-neg2 91.4%

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

      3. sub-neg 91.4%

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

      4. associate-+l+ 91.4%

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

      5. fma-define 91.5%

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

      6. sub-neg 91.5%

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

      7. metadata-eval 91.5%

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

      8. +-commutative 91.5%

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

      9. unsub-neg 91.5%

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

      10. distribute-frac-neg2 91.5%

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

      11. remove-double-neg 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in z around 0 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-110}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) - \frac{0.0027777777777778}{x}, \log x \cdot \left(x + -0.5\right)\right)\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) - \frac{0.0027777777777778}{x}, \log x \cdot \left(x + -0.5\right)\right)\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.6e-110)
   (/
    (+
     0.083333333333333
     (* z (* z (+ 0.0007936500793651 (- y (/ 0.0027777777777778 z))))))
    x)
   (-
    (+
     0.91893853320467
     (+
      (/ 0.083333333333333 x)
      (fma
       z
       (- (* z (+ (/ 0.0007936500793651 x) (/ y x))) (/ 0.0027777777777778 x))
       (* (log x) (+ x -0.5)))))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.6e-110) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + fma(z, ((z * ((0.0007936500793651 / x) + (y / x))) - (0.0027777777777778 / x)), (log(x) * (x + -0.5))))) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.6e-110)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(z * Float64(0.0007936500793651 + Float64(y - Float64(0.0027777777777778 / z)))))) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 / x) + fma(z, Float64(Float64(z * Float64(Float64(0.0007936500793651 / x) + Float64(y / x))) - Float64(0.0027777777777778 / x)), Float64(log(x) * Float64(x + -0.5))))) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.6e-110], N[(N[(0.083333333333333 + N[(z * N[(z * N[(0.0007936500793651 + N[(y - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(z * N[(N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e-110

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. distribute-frac-neg2 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. unsub-neg 99.7%

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

      10. distribute-frac-neg2 99.7%

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

      11. remove-double-neg 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in z around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   if 4.6000000000000003e-110 < x

   1. Initial program 91.4%

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

      1. remove-double-neg 91.4%

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

      2. distribute-frac-neg2 91.4%

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

      3. sub-neg 91.4%

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

      4. associate-+l+ 91.4%

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

      5. fma-define 91.5%

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

      6. sub-neg 91.5%

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

      7. metadata-eval 91.5%

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

      8. +-commutative 91.5%

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

      9. unsub-neg 91.5%

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

      10. distribute-frac-neg2 91.5%

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

      11. remove-double-neg 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in z around 0 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) - \frac{0.0027777777777778}{x}, \log x \cdot \left(x + -0.5\right)\right)\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-40}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + \left(x - 0.5\right) \cdot \log x\right)\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e-40)
   (/
    (+
     0.083333333333333
     (* z (* z (+ 0.0007936500793651 (- y (/ 0.0027777777777778 z))))))
    x)
   (-
    (+
     0.91893853320467
     (+
      (/ 0.083333333333333 x)
      (+ (* z (* (+ 0.0007936500793651 y) (/ z x))) (* (- x 0.5) (log x)))))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-40) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + ((x - 0.5) * log(x))))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1d-40) then
        tmp = (0.083333333333333d0 + (z * (z * (0.0007936500793651d0 + (y - (0.0027777777777778d0 / z)))))) / x
    else
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 / x) + ((z * ((0.0007936500793651d0 + y) * (z / x))) + ((x - 0.5d0) * log(x))))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-40) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + ((x - 0.5) * Math.log(x))))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1e-40:
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x
	else:
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + ((x - 0.5) * math.log(x))))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e-40)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(z * Float64(0.0007936500793651 + Float64(y - Float64(0.0027777777777778 / z)))))) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 / x) + Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * Float64(z / x))) + Float64(Float64(x - 0.5) * log(x))))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1e-40)
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	else
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + ((x - 0.5) * log(x))))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e-40], N[(N[(0.083333333333333 + N[(z * N[(z * N[(0.0007936500793651 + N[(y - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999993e-41

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. distribute-frac-neg2 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. unsub-neg 99.7%

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

      10. distribute-frac-neg2 99.7%

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

      11. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in z around inf 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   if 9.9999999999999993e-41 < x

   1. Initial program 90.0%

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

      1. remove-double-neg 90.0%

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

      2. distribute-frac-neg2 90.0%

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

      3. sub-neg 90.0%

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

      4. associate-+l+ 89.9%

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

      5. fma-define 90.1%

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

      6. sub-neg 90.1%

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

      7. metadata-eval 90.1%

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

      8. +-commutative 90.1%

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

      9. unsub-neg 90.1%

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

      10. distribute-frac-neg2 90.1%

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

      11. remove-double-neg 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in z around 0 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around 0 99.5%

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

   9. Taylor expanded in z around inf 94.5%

      \[\leadsto \left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(\color{blue}{{z}^{2} \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right)} + \log x \cdot \left(x - 0.5\right)\right)\right)\right) - x \]
   10. Step-by-step derivation

       1. unpow2 94.5%

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

       2. associate-*r/ 94.5%

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

       3. metadata-eval 94.5%

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

       4. associate-*l* 99.5%

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

       5. distribute-rgt-in 99.5%

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

       6. associate-*l/ 99.5%

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

       7. associate-*r/ 99.5%

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

       8. associate-*l/ 95.0%

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

       9. associate-/l* 99.5%

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

       10. distribute-rgt-out 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-40}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + \left(x - 0.5\right) \cdot \log x\right)\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14800000000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + x \cdot \log x\right)\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 14800000000.0)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778)))
     x))
   (-
    (+
     0.91893853320467
     (+
      (/ 0.083333333333333 x)
      (+ (* z (* (+ 0.0007936500793651 y) (/ z x))) (* x (log x)))))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 14800000000.0) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * log(x))))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 14800000000.0d0) then
        tmp = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((0.083333333333333d0 + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0))) / x)
    else
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 / x) + ((z * ((0.0007936500793651d0 + y) * (z / x))) + (x * log(x))))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 14800000000.0) {
		tmp = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * Math.log(x))))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 14800000000.0:
		tmp = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x)
	else:
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * math.log(x))))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 14800000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 / x) + Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * Float64(z / x))) + Float64(x * log(x))))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 14800000000.0)
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778))) / x);
	else
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * log(x))))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 14800000000.0], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.48e10

   1. Initial program 99.7%

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

   if 1.48e10 < x

   1. Initial program 88.0%

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

      1. remove-double-neg 88.0%

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

      2. distribute-frac-neg2 88.0%

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

      3. sub-neg 88.0%

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

      4. associate-+l+ 88.0%

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

      5. fma-define 88.1%

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

      6. sub-neg 88.1%

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

      7. metadata-eval 88.1%

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

      8. +-commutative 88.1%

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

      9. unsub-neg 88.1%

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

      10. distribute-frac-neg2 88.1%

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

      11. remove-double-neg 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in z around 0 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in z around 0 99.4%

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

   9. Taylor expanded in z around inf 93.4%

      \[\leadsto \left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(\color{blue}{{z}^{2} \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right)} + \log x \cdot \left(x - 0.5\right)\right)\right)\right) - x \]
   10. Step-by-step derivation

       1. unpow2 93.4%

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

       2. associate-*r/ 93.4%

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

       3. metadata-eval 93.4%

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

       4. associate-*l* 99.4%

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

       5. distribute-rgt-in 99.4%

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

       6. associate-*l/ 99.4%

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

       7. associate-*r/ 99.5%

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

       8. associate-*l/ 94.0%

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

       9. associate-/l* 99.4%

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

       10. distribute-rgt-out 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in x around inf 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 14800000000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + x \cdot \log x\right)\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0039:\\ \;\;\;\;\frac{0.083333333333333 + \left(x \cdot \left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + x \cdot \log x\right)\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0039)
   (/
    (+
     0.083333333333333
     (+
      (* x (+ 0.91893853320467 (* (log x) -0.5)))
      (* z (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778))))
    x)
   (-
    (+
     0.91893853320467
     (+
      (/ 0.083333333333333 x)
      (+ (* z (* (+ 0.0007936500793651 y) (/ z x))) (* x (log x)))))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0039) {
		tmp = (0.083333333333333 + ((x * (0.91893853320467 + (log(x) * -0.5))) + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * log(x))))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.0039d0) then
        tmp = (0.083333333333333d0 + ((x * (0.91893853320467d0 + (log(x) * (-0.5d0)))) + (z * ((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0)))) / x
    else
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 / x) + ((z * ((0.0007936500793651d0 + y) * (z / x))) + (x * log(x))))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0039) {
		tmp = (0.083333333333333 + ((x * (0.91893853320467 + (Math.log(x) * -0.5))) + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * Math.log(x))))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0039:
		tmp = (0.083333333333333 + ((x * (0.91893853320467 + (math.log(x) * -0.5))) + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)))) / x
	else:
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * math.log(x))))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0039)
		tmp = Float64(Float64(0.083333333333333 + Float64(Float64(x * Float64(0.91893853320467 + Float64(log(x) * -0.5))) + Float64(z * Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778)))) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 / x) + Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * Float64(z / x))) + Float64(x * log(x))))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0039)
		tmp = (0.083333333333333 + ((x * (0.91893853320467 + (log(x) * -0.5))) + (z * ((z * (0.0007936500793651 + y)) - 0.0027777777777778)))) / x;
	else
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * log(x))))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0039], N[(N[(0.083333333333333 + N[(N[(x * N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0038999999999999998

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. distribute-frac-neg2 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. unsub-neg 99.7%

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

      10. distribute-frac-neg2 99.7%

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

      11. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.4%

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

   if 0.0038999999999999998 < x

   1. Initial program 88.7%

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

      1. remove-double-neg 88.7%

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

      2. distribute-frac-neg2 88.7%

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

      3. sub-neg 88.7%

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

      4. associate-+l+ 88.7%

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

      5. fma-define 88.8%

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

      6. sub-neg 88.8%

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

      7. metadata-eval 88.8%

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

      8. +-commutative 88.8%

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

      9. unsub-neg 88.8%

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

      10. distribute-frac-neg2 88.8%

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

      11. remove-double-neg 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in z around 0 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in z around 0 99.4%

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

   9. Taylor expanded in z around inf 93.8%

      \[\leadsto \left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(\color{blue}{{z}^{2} \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right)} + \log x \cdot \left(x - 0.5\right)\right)\right)\right) - x \]
   10. Step-by-step derivation

       1. unpow2 93.8%

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

       2. associate-*r/ 93.8%

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

       3. metadata-eval 93.8%

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

       4. associate-*l* 99.4%

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

       5. distribute-rgt-in 99.4%

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

       6. associate-*l/ 99.4%

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

       7. associate-*r/ 99.5%

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

       8. associate-*l/ 94.3%

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

       9. associate-/l* 99.4%

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

       10. distribute-rgt-out 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in x around inf 98.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0039:\\ \;\;\;\;\frac{0.083333333333333 + \left(x \cdot \left(0.91893853320467 + \log x \cdot -0.5\right) + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + x \cdot \log x\right)\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-43}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + x \cdot \log x\right)\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e-43)
   (/
    (+
     0.083333333333333
     (* z (* z (+ 0.0007936500793651 (- y (/ 0.0027777777777778 z))))))
    x)
   (-
    (+
     0.91893853320467
     (+
      (/ 0.083333333333333 x)
      (+ (* z (* (+ 0.0007936500793651 y) (/ z x))) (* x (log x)))))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-43) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * log(x))))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1d-43) then
        tmp = (0.083333333333333d0 + (z * (z * (0.0007936500793651d0 + (y - (0.0027777777777778d0 / z)))))) / x
    else
        tmp = (0.91893853320467d0 + ((0.083333333333333d0 / x) + ((z * ((0.0007936500793651d0 + y) * (z / x))) + (x * log(x))))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e-43) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * Math.log(x))))) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1e-43:
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x
	else:
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * math.log(x))))) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e-43)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(z * Float64(0.0007936500793651 + Float64(y - Float64(0.0027777777777778 / z)))))) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(0.083333333333333 / x) + Float64(Float64(z * Float64(Float64(0.0007936500793651 + y) * Float64(z / x))) + Float64(x * log(x))))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1e-43)
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	else
		tmp = (0.91893853320467 + ((0.083333333333333 / x) + ((z * ((0.0007936500793651 + y) * (z / x))) + (x * log(x))))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e-43], N[(N[(0.083333333333333 + N[(z * N[(z * N[(0.0007936500793651 + N[(y - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(z * N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000008e-43

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. distribute-frac-neg2 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. unsub-neg 99.7%

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

      10. distribute-frac-neg2 99.7%

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

      11. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in z around inf 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   if 1.00000000000000008e-43 < x

   1. Initial program 90.0%

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

      1. remove-double-neg 90.0%

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

      2. distribute-frac-neg2 90.0%

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

      3. sub-neg 90.0%

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

      4. associate-+l+ 89.9%

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

      5. fma-define 90.1%

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

      6. sub-neg 90.1%

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

      7. metadata-eval 90.1%

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

      8. +-commutative 90.1%

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

      9. unsub-neg 90.1%

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

      10. distribute-frac-neg2 90.1%

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

      11. remove-double-neg 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in z around 0 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around 0 99.5%

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

   9. Taylor expanded in z around inf 94.5%

      \[\leadsto \left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(\color{blue}{{z}^{2} \cdot \left(0.0007936500793651 \cdot \frac{1}{x} + \frac{y}{x}\right)} + \log x \cdot \left(x - 0.5\right)\right)\right)\right) - x \]
   10. Step-by-step derivation

       1. unpow2 94.5%

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

       2. associate-*r/ 94.5%

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

       3. metadata-eval 94.5%

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

       4. associate-*l* 99.5%

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

       5. distribute-rgt-in 99.5%

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

       6. associate-*l/ 99.5%

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

       7. associate-*r/ 99.5%

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

       8. associate-*l/ 95.0%

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

       9. associate-/l* 99.5%

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

       10. distribute-rgt-out 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in x around inf 97.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-43}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\frac{0.083333333333333}{x} + \left(z \cdot \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) + x \cdot \log x\right)\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + \left(y - \frac{0.0027777777777778}{z}\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.4e+51)
   (/
    (+
     0.083333333333333
     (* z (* z (+ 0.0007936500793651 (- y (/ 0.0027777777777778 z))))))
    x)
   (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.4e+51) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = x * (log(x) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.4d+51) then
        tmp = (0.083333333333333d0 + (z * (z * (0.0007936500793651d0 + (y - (0.0027777777777778d0 / z)))))) / x
    else
        tmp = x * (log(x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.4e+51) {
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	} else {
		tmp = x * (Math.log(x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.4e+51:
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x
	else:
		tmp = x * (math.log(x) + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.4e+51)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(z * Float64(0.0007936500793651 + Float64(y - Float64(0.0027777777777778 / z)))))) / x);
	else
		tmp = Float64(x * Float64(log(x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.4e+51)
		tmp = (0.083333333333333 + (z * (z * (0.0007936500793651 + (y - (0.0027777777777778 / z)))))) / x;
	else
		tmp = x * (log(x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.4e+51], N[(N[(0.083333333333333 + N[(z * N[(z * N[(0.0007936500793651 + N[(y - N[(0.0027777777777778 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999984e51

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. distribute-frac-neg2 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. unsub-neg 99.7%

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

      10. distribute-frac-neg2 99.7%

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

      11. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 92.6%

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

   6. Taylor expanded in z around inf 92.6%

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

      1. associate--l+ 92.6%

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

      2. associate-*r/ 92.6%

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

      3. metadata-eval 92.6%

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

   8. Simplified 92.6%

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

   if 3.39999999999999984e51 < x

   1. Initial program 86.4%

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

      1. remove-double-neg 86.4%

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

      2. distribute-frac-neg2 86.4%

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

      3. sub-neg 86.4%

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

      4. associate-+l+ 86.4%

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

      5. fma-define 86.5%

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

      6. sub-neg 86.5%

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

      7. metadata-eval 86.5%

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

      8. +-commutative 86.5%

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

      9. unsub-neg 86.5%

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

      10. distribute-frac-neg2 86.5%

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

      11. remove-double-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in x around inf 84.1%

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

      1. sub-neg 84.1%

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

      2. mul-1-neg 84.1%

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

      3. log-rec 84.1%

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

      4. remove-double-neg 84.1%

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

      5. metadata-eval 84.1%

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

   7. Simplified 84.1%

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

Alternative 8: 62.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0008 \lor \neg \left(y \leq 0.00067\right):\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot y - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot 0.0007936500793651 - 0.0027777777777778\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0008) (not (<= y 0.00067)))
   (/ (+ 0.083333333333333 (* z (- (* z y) 0.0027777777777778))) x)
   (/
    (+ 0.083333333333333 (* z (- (* z 0.0007936500793651) 0.0027777777777778)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0008) || !(y <= 0.00067)) {
		tmp = (0.083333333333333 + (z * ((z * y) - 0.0027777777777778))) / x;
	} else {
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / 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 ((y <= (-0.0008d0)) .or. (.not. (y <= 0.00067d0))) then
        tmp = (0.083333333333333d0 + (z * ((z * y) - 0.0027777777777778d0))) / x
    else
        tmp = (0.083333333333333d0 + (z * ((z * 0.0007936500793651d0) - 0.0027777777777778d0))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0008) || !(y <= 0.00067)) {
		tmp = (0.083333333333333 + (z * ((z * y) - 0.0027777777777778))) / x;
	} else {
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0008) or not (y <= 0.00067):
		tmp = (0.083333333333333 + (z * ((z * y) - 0.0027777777777778))) / x
	else:
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0008) || !(y <= 0.00067))
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * y) - 0.0027777777777778))) / x);
	else
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * 0.0007936500793651) - 0.0027777777777778))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0008) || ~((y <= 0.00067)))
		tmp = (0.083333333333333 + (z * ((z * y) - 0.0027777777777778))) / x;
	else
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0008], N[Not[LessEqual[y, 0.00067]], $MachinePrecision]], N[(N[(0.083333333333333 + N[(z * N[(N[(z * y), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.083333333333333 + N[(z * N[(N[(z * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e-4 or 6.7000000000000002e-4 < y

   1. Initial program 93.1%

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

      1. remove-double-neg 93.1%

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

      2. distribute-frac-neg2 93.1%

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

      3. sub-neg 93.1%

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

      4. associate-+l+ 93.1%

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

      5. fma-define 93.2%

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

      6. sub-neg 93.2%

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

      7. metadata-eval 93.2%

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

      8. +-commutative 93.2%

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

      9. unsub-neg 93.2%

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

      10. distribute-frac-neg2 93.2%

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

      11. remove-double-neg 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 65.3%

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

   6. Taylor expanded in y around inf 64.8%

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

      1. *-commutative 64.8%

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

   8. Simplified 64.8%

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

   if -8.00000000000000038e-4 < y < 6.7000000000000002e-4

   1. Initial program 95.2%

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

      1. remove-double-neg 95.2%

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

      2. distribute-frac-neg2 95.2%

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

      3. sub-neg 95.2%

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

      4. associate-+l+ 95.2%

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

      5. fma-define 95.2%

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

      6. sub-neg 95.2%

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

      7. metadata-eval 95.2%

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

      8. +-commutative 95.2%

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

      9. unsub-neg 95.2%

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

      10. distribute-frac-neg2 95.2%

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

      11. remove-double-neg 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 55.1%

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

   6. Taylor expanded in y around 0 54.6%

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

      1. *-commutative 54.6%

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

   8. Simplified 54.6%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0008 \lor \neg \left(y \leq 0.00067\right):\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot y - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot 0.0007936500793651 - 0.0027777777777778\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-22} \lor \neg \left(z \leq 2.15 \cdot 10^{-10}\right):\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-22) (not (<= z 2.15e-10)))
   (* y (/ (* z z) x))
   (/ 1.0 (* x 12.000000000000048))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-22) || !(z <= 2.15e-10)) {
		tmp = y * ((z * z) / x);
	} else {
		tmp = 1.0 / (x * 12.000000000000048);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8d-22)) .or. (.not. (z <= 2.15d-10))) then
        tmp = y * ((z * z) / x)
    else
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-22) || !(z <= 2.15e-10)) {
		tmp = y * ((z * z) / x);
	} else {
		tmp = 1.0 / (x * 12.000000000000048);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e-22) or not (z <= 2.15e-10):
		tmp = y * ((z * z) / x)
	else:
		tmp = 1.0 / (x * 12.000000000000048)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-22) || !(z <= 2.15e-10))
		tmp = Float64(y * Float64(Float64(z * z) / x));
	else
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8e-22) || ~((z <= 2.15e-10)))
		tmp = y * ((z * z) / x);
	else
		tmp = 1.0 / (x * 12.000000000000048);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e-22], N[Not[LessEqual[z, 2.15e-10]], $MachinePrecision]], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.0000000000000004e-22 or 2.15000000000000007e-10 < z

   1. Initial program 89.3%

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

      1. remove-double-neg 89.3%

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

      2. distribute-frac-neg2 89.3%

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

      3. sub-neg 89.3%

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

      4. associate-+l+ 89.3%

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

      5. fma-define 89.3%

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

      6. sub-neg 89.3%

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

      7. metadata-eval 89.3%

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

      8. +-commutative 89.3%

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

      9. unsub-neg 89.3%

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

      10. distribute-frac-neg2 89.3%

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

      11. remove-double-neg 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around inf 48.2%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

      1. unpow2 50.3%

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

   9. Applied egg-rr 50.3%

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

   if -8.0000000000000004e-22 < z < 2.15000000000000007e-10

   1. Initial program 99.5%

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

      1. remove-double-neg 99.5%

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

      2. distribute-frac-neg2 99.5%

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

      3. sub-neg 99.5%

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

      4. associate-+l+ 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. unsub-neg 99.6%

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

      10. distribute-frac-neg2 99.6%

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

      11. remove-double-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 45.3%

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

   6. Taylor expanded in z around 0 42.9%

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

      1. div-inv 42.9%

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

      2. *-commutative 42.9%

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

   8. Applied egg-rr 42.9%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
   9. Step-by-step derivation

      1. associate-*l/ 42.9%

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

      2. metadata-eval 42.9%

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

      3. clear-num 42.9%

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

      4. div-inv 43.0%

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

      5. metadata-eval 43.0%

         \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   10. Applied egg-rr 43.0%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-22} \lor \neg \left(z \leq 2.15 \cdot 10^{-10}\right):\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1350000:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot 0.0007936500793651 - 0.0027777777777778\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1350000.0)
   (* y (/ (* z z) x))
   (/
    (+ 0.083333333333333 (* z (- (* z 0.0007936500793651) 0.0027777777777778)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1350000.0) {
		tmp = y * ((z * z) / x);
	} else {
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / 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 (y <= (-1350000.0d0)) then
        tmp = y * ((z * z) / x)
    else
        tmp = (0.083333333333333d0 + (z * ((z * 0.0007936500793651d0) - 0.0027777777777778d0))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1350000.0) {
		tmp = y * ((z * z) / x);
	} else {
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1350000.0:
		tmp = y * ((z * z) / x)
	else:
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1350000.0)
		tmp = Float64(y * Float64(Float64(z * z) / x));
	else
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * 0.0007936500793651) - 0.0027777777777778))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1350000.0)
		tmp = y * ((z * z) / x);
	else
		tmp = (0.083333333333333 + (z * ((z * 0.0007936500793651) - 0.0027777777777778))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1350000.0], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.083333333333333 + N[(z * N[(N[(z * 0.0007936500793651), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e6

   1. Initial program 92.4%

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

      1. remove-double-neg 92.4%

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

      2. distribute-frac-neg2 92.4%

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

      3. sub-neg 92.4%

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

      4. associate-+l+ 92.4%

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

      5. fma-define 92.6%

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

      6. sub-neg 92.6%

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

      7. metadata-eval 92.6%

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

      8. +-commutative 92.6%

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

      9. unsub-neg 92.6%

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

      10. distribute-frac-neg2 92.6%

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

      11. remove-double-neg 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in y around inf 52.3%

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

      1. associate-/l* 53.6%

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

   7. Simplified 53.6%

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

      1. unpow2 53.6%

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

   9. Applied egg-rr 53.6%

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

   if -1.35e6 < y

   1. Initial program 94.7%

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

      1. remove-double-neg 94.7%

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

      2. distribute-frac-neg2 94.7%

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

      3. sub-neg 94.7%

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

      4. associate-+l+ 94.7%

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

      5. fma-define 94.8%

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

      6. sub-neg 94.8%

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

      7. metadata-eval 94.8%

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

      8. +-commutative 94.8%

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

      9. unsub-neg 94.8%

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

      10. distribute-frac-neg2 94.8%

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

      11. remove-double-neg 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in x around 0 58.5%

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

   6. Taylor expanded in y around 0 54.2%

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

      1. *-commutative 54.2%

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

   8. Simplified 54.2%

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

Alternative 11: 62.5% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. remove-double-neg 94.1%

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

   2. distribute-frac-neg2 94.1%

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

   3. sub-neg 94.1%

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

   4. associate-+l+ 94.1%

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

   5. fma-define 94.2%

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

   6. sub-neg 94.2%

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

   7. metadata-eval 94.2%

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

   8. +-commutative 94.2%

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

   9. unsub-neg 94.2%

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

   10. distribute-frac-neg2 94.2%

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

   11. remove-double-neg 94.2%

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

3. Simplified 94.2%

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

5. Taylor expanded in x around 0 60.3%

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

6. Taylor expanded in z around inf 60.3%

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

   1. associate--l+ 60.3%

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

   2. associate-*r/ 60.3%

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

   3. metadata-eval 60.3%

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

8. Simplified 60.3%

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

Alternative 12: 62.5% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. remove-double-neg 94.1%

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

   2. distribute-frac-neg2 94.1%

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

   3. sub-neg 94.1%

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

   4. associate-+l+ 94.1%

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

   5. fma-define 94.2%

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

   6. sub-neg 94.2%

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

   7. metadata-eval 94.2%

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

   8. +-commutative 94.2%

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

   9. unsub-neg 94.2%

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

   10. distribute-frac-neg2 94.2%

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

   11. remove-double-neg 94.2%

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

3. Simplified 94.2%

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

5. Taylor expanded in x around 0 60.3%

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

Alternative 13: 23.7% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 12.000000000000048} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (* x 12.000000000000048)))

C

double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (x * 12.000000000000048d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Python

def code(x, y, z):
	return 1.0 / (x * 12.000000000000048)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(x * 12.000000000000048))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (x * 12.000000000000048);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. remove-double-neg 94.1%

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

   2. distribute-frac-neg2 94.1%

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

   3. sub-neg 94.1%

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

   4. associate-+l+ 94.1%

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

   5. fma-define 94.2%

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

   6. sub-neg 94.2%

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

   7. metadata-eval 94.2%

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

   8. +-commutative 94.2%

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

   9. unsub-neg 94.2%

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

   10. distribute-frac-neg2 94.2%

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

   11. remove-double-neg 94.2%

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

3. Simplified 94.2%

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

5. Taylor expanded in x around 0 60.3%

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

6. Taylor expanded in z around 0 22.1%

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

   1. div-inv 22.1%

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

   2. *-commutative 22.1%

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

8. Applied egg-rr 22.1%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]
9. Step-by-step derivation

   1. associate-*l/ 22.1%

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

   2. metadata-eval 22.1%

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

   3. clear-num 22.1%

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

   4. div-inv 22.1%

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

   5. metadata-eval 22.1%

      \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

10. Applied egg-rr 22.1%

    \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
11. Add Preprocessing

Alternative 14: 23.6% 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 94.1%

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

   1. remove-double-neg 94.1%

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

   2. distribute-frac-neg2 94.1%

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

   3. sub-neg 94.1%

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

   4. associate-+l+ 94.1%

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

   5. fma-define 94.2%

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

   6. sub-neg 94.2%

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

   7. metadata-eval 94.2%

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

   8. +-commutative 94.2%

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

   9. unsub-neg 94.2%

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

   10. distribute-frac-neg2 94.2%

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

   11. remove-double-neg 94.2%

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

3. Simplified 94.2%

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

5. Taylor expanded in x around 0 60.3%

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

6. Taylor expanded in z around 0 22.1%

   \[\leadsto \frac{\color{blue}{0.083333333333333}}{x} \]
7. Add Preprocessing

Developer Target 1: 98.5% 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 2024176 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

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