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

Percentage Accurate: 93.6% → 99.6%

Time: 12.4s

Alternatives: 16

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq 0.000793650079365 \lor \neg \left(y + 0.0007936500793651 \leq 0.0007937\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\frac{z}{x} \cdot \left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{y} + z\right), y, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, \frac{0.083333333333333}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (+ y 0.0007936500793651) 0.000793650079365)
         (not (<= (+ y 0.0007936500793651) 0.0007937)))
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (fma
     (* (/ z x) (+ (/ (fma 0.0007936500793651 z -0.0027777777777778) y) z))
     y
     (/ 0.083333333333333 x)))
   (+
    (- 0.91893853320467 x)
    (fma
     (log x)
     (+ -0.5 x)
     (fma
      z
      (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
      (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y + 0.0007936500793651) <= 0.000793650079365) || !((y + 0.0007936500793651) <= 0.0007937)) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(((z / x) * ((fma(0.0007936500793651, z, -0.0027777777777778) / y) + z)), y, (0.083333333333333 / x));
	} else {
		tmp = (0.91893853320467 - x) + fma(log(x), (-0.5 + x), fma(z, (fma(0.0007936500793651, z, -0.0027777777777778) / x), (0.083333333333333 / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(y + 0.0007936500793651) <= 0.000793650079365) || !(Float64(y + 0.0007936500793651) <= 0.0007937))
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + fma(Float64(Float64(z / x) * Float64(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / y) + z)), y, Float64(0.083333333333333 / x)));
	else
		tmp = Float64(Float64(0.91893853320467 - x) + fma(log(x), Float64(-0.5 + x), fma(z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), Float64(0.083333333333333 / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.000793650079365], N[Not[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.0007937]], $MachinePrecision]], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z / x), $MachinePrecision] * N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(-0.5 + x), $MachinePrecision] + N[(z * N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.93650079365000051e-4 or 7.9370000000000005e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 93.3%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 99.7%

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

   if 7.93650079365000051e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.9370000000000005e-4

   1. Initial program 95.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.9

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq 0.000793650079365 \lor \neg \left(y + 0.0007936500793651 \leq 0.0007937\right):\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(\frac{z}{x} \cdot \left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{y} + z\right), y, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, \frac{0.083333333333333}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right)\\ \mathbf{if}\;x \leq 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(t\_0, z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-\log x}{x}, 0.5, \mathsf{fma}\left(\frac{t\_0}{x}, \frac{z}{x}, \frac{0.91893853320467}{x}\right) + \frac{0.083333333333333}{x \cdot x}\right) + \log x, x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ 0.0007936500793651 y) z -0.0027777777777778)))
   (if (<= x 1e-8)
     (/
      (fma
       (+ (fma (- x 0.5) (log x) (- x)) 0.91893853320467)
       x
       (fma t_0 z 0.083333333333333))
      x)
     (fma
      (+
       (fma
        (/ (- (log x)) x)
        0.5
        (+
         (fma (/ t_0 x) (/ z x) (/ 0.91893853320467 x))
         (/ 0.083333333333333 (* x x))))
       (log x))
      x
      (- x)))))

C

double code(double x, double y, double z) {
	double t_0 = fma((0.0007936500793651 + y), z, -0.0027777777777778);
	double tmp;
	if (x <= 1e-8) {
		tmp = fma((fma((x - 0.5), log(x), -x) + 0.91893853320467), x, fma(t_0, z, 0.083333333333333)) / x;
	} else {
		tmp = fma((fma((-log(x) / x), 0.5, (fma((t_0 / x), (z / x), (0.91893853320467 / x)) + (0.083333333333333 / (x * x)))) + log(x)), x, -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778)
	tmp = 0.0
	if (x <= 1e-8)
		tmp = Float64(fma(Float64(fma(Float64(x - 0.5), log(x), Float64(-x)) + 0.91893853320467), x, fma(t_0, z, 0.083333333333333)) / x);
	else
		tmp = fma(Float64(fma(Float64(Float64(-log(x)) / x), 0.5, Float64(fma(Float64(t_0 / x), Float64(z / x), Float64(0.91893853320467 / x)) + Float64(0.083333333333333 / Float64(x * x)))) + log(x)), x, Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision]}, If[LessEqual[x, 1e-8], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(t$95$0 * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[((-N[Log[x], $MachinePrecision]) / x), $MachinePrecision] * 0.5 + N[(N[(N[(t$95$0 / x), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.91893853320467 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] * x + (-x)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.7%

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

   if 1e-8 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

Alternative 3: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
       (/
        (+
         (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
         0.083333333333333)
        x))
      -1e+144)
   (* (* (/ z x) y) z)
   (/
    (fma (fma z 0.0007936500793651 -0.0027777777777778) z 0.083333333333333)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)) <= -1e+144) {
		tmp = ((z / x) * y) * z;
	} else {
		tmp = fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (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)) <= -1e+144)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	else
		tmp = Float64(fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[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], -1e+144], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -1.00000000000000002e144

   1. Initial program 88.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{x} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 94.0%

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

   if -1.00000000000000002e144 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 90.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.5%

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

Alternative 4: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -1e+101)
     (* (* (/ z x) y) z)
     (if (<= t_0 2e-33)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (/
        (fma
         (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
         z
         0.083333333333333)
        x)))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -1e+101) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 2e-33) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -1e+101)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 2e-33)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+101], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e-33], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -9.9999999999999998e100

   1. Initial program 89.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{x} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 88.6%

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

   if -9.9999999999999998e100 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.0000000000000001e-33

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 2.0000000000000001e-33 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 80.3

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

   5. Applied rewrites 80.3%

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

Alternative 5: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\left(-x\right) + \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -1e+101)
     (* (* (/ z x) y) z)
     (if (<= t_0 2e-33)
       (+ (- x) (fma (log x) (- x 0.5) (/ 0.083333333333333 x)))
       (/
        (fma
         (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
         z
         0.083333333333333)
        x)))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -1e+101) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 2e-33) {
		tmp = -x + fma(log(x), (x - 0.5), (0.083333333333333 / x));
	} else {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -1e+101)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 2e-33)
		tmp = Float64(Float64(-x) + fma(log(x), Float64(x - 0.5), Float64(0.083333333333333 / x)));
	else
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+101], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e-33], N[((-x) + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -9.9999999999999998e100

   1. Initial program 89.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{x} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 88.6%

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

   if -9.9999999999999998e100 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.0000000000000001e-33

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 96.7%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 96.6%

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

   if 2.0000000000000001e-33 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 80.3

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

   5. Applied rewrites 80.3%

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

Alternative 6: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) + \mathsf{fma}\left(\log x, -0.5 + x, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, \frac{0.083333333333333}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.2e+183)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
      0.083333333333333)
     x))
   (+
    (- x)
    (fma
     (log x)
     (+ -0.5 x)
     (fma
      z
      (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
      (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.2e+183) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	} else {
		tmp = -x + fma(log(x), (-0.5 + x), fma(z, (fma(0.0007936500793651, z, -0.0027777777777778) / x), (0.083333333333333 / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.2e+183)
		tmp = 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));
	else
		tmp = Float64(Float64(-x) + fma(log(x), Float64(-0.5 + x), fma(z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), Float64(0.083333333333333 / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.2e+183], 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], N[((-x) + N[(N[Log[x], $MachinePrecision] * N[(-0.5 + x), $MachinePrecision] + N[(z * N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1999999999999999e183

   1. Initial program 97.9%

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

   if 5.1999999999999999e183 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 18.6

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 91.2%

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

Alternative 7: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 100000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) + \frac{\left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 100000000.0)
   (/
    (fma
     (+ (fma (- x 0.5) (log x) (- x)) 0.91893853320467)
     x
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (fma
    (- x 0.5)
    (log x)
    (+ (- 0.91893853320467 x) (/ (* (+ 0.0007936500793651 y) (* z z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 100000000.0) {
		tmp = fma((fma((x - 0.5), log(x), -x) + 0.91893853320467), x, fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) + (((0.0007936500793651 + y) * (z * z)) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 100000000.0)
		tmp = Float64(fma(Float64(fma(Float64(x - 0.5), log(x), Float64(-x)) + 0.91893853320467), x, fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) + Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z * z)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 100000000.0], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.7%

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

   if 1e8 < x

   1. Initial program 87.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 87.7

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 87.9

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

   7. Applied rewrites 87.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

   9. Applied rewrites 87.9%

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

Alternative 8: 93.6% 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]

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. Add Preprocessing
3. Add Preprocessing

Alternative 9: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 1 < x

   1. Initial program 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 88.1

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 88.1

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

   7. Applied rewrites 88.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

   9. Applied rewrites 88.1%

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

Alternative 10: 83.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.3e+20)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.3e+20) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.3e+20)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.3e+20], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3e20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 2.3e20 < x

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lower-pow.f64 87.1

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 70.9

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

   7. Applied rewrites 70.9%

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

Alternative 11: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+101)
     (* (* (/ z x) y) z)
     (if (<= t_0 1000000.0)
       (/
        (fma
         (fma z 0.0007936500793651 -0.0027777777777778)
         z
         0.083333333333333)
        x)
       (* (* (/ z x) (+ 0.0007936500793651 y)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+101) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 1000000.0) {
		tmp = fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * (0.0007936500793651 + y)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+101)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 1000000.0)
		tmp = Float64(fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+101], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 1000000.0], N[(N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -9.9999999999999998e100

   1. Initial program 89.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{x} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 88.6%

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

   if -9.9999999999999998e100 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1e6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.4%

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

   if 1e6 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 90.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.5%

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

Alternative 12: 44.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -200:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.001:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y 0.0007936500793651) -200.0)
   (* (* (/ z x) y) z)
   (if (<= (+ y 0.0007936500793651) 0.001)
     (* (* (/ z x) 0.0007936500793651) z)
     (* (* z y) (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -200.0) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 0.001) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = (z * y) * (z / 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 + 0.0007936500793651d0) <= (-200.0d0)) then
        tmp = ((z / x) * y) * z
    else if ((y + 0.0007936500793651d0) <= 0.001d0) then
        tmp = ((z / x) * 0.0007936500793651d0) * z
    else
        tmp = (z * y) * (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -200.0) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 0.001) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = (z * y) * (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y + 0.0007936500793651) <= -200.0:
		tmp = ((z / x) * y) * z
	elif (y + 0.0007936500793651) <= 0.001:
		tmp = ((z / x) * 0.0007936500793651) * z
	else:
		tmp = (z * y) * (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -200.0)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (Float64(y + 0.0007936500793651) <= 0.001)
		tmp = Float64(Float64(Float64(z / x) * 0.0007936500793651) * z);
	else
		tmp = Float64(Float64(z * y) * Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -200.0)
		tmp = ((z / x) * y) * z;
	elseif ((y + 0.0007936500793651) <= 0.001)
		tmp = ((z / x) * 0.0007936500793651) * z;
	else
		tmp = (z * y) * (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -200.0], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.001], N[(N[(N[(z / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -200

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{x} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

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

   if -200 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 1e-3

   1. Initial program 94.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 40.5%

      \[\leadsto \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \]

   if 1e-3 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 55.1%

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

   9. Applied rewrites 55.1%

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

Alternative 13: 44.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -200:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 1000:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y 0.0007936500793651) -200.0)
   (* (* (/ z x) y) z)
   (if (<= (+ y 0.0007936500793651) 1000.0)
     (* (* (/ z x) 0.0007936500793651) z)
     (* y (/ (* z z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -200.0) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 1000.0) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = y * ((z * z) / 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 + 0.0007936500793651d0) <= (-200.0d0)) then
        tmp = ((z / x) * y) * z
    else if ((y + 0.0007936500793651d0) <= 1000.0d0) then
        tmp = ((z / x) * 0.0007936500793651d0) * z
    else
        tmp = y * ((z * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -200.0) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 1000.0) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y + 0.0007936500793651) <= -200.0:
		tmp = ((z / x) * y) * z
	elif (y + 0.0007936500793651) <= 1000.0:
		tmp = ((z / x) * 0.0007936500793651) * z
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -200.0)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (Float64(y + 0.0007936500793651) <= 1000.0)
		tmp = Float64(Float64(Float64(z / x) * 0.0007936500793651) * z);
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -200.0)
		tmp = ((z / x) * y) * z;
	elseif ((y + 0.0007936500793651) <= 1000.0)
		tmp = ((z / x) * 0.0007936500793651) * z;
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -200.0], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 1000.0], N[(N[(N[(z / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -200

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{x} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

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

   if -200 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 1e3

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

      \[\leadsto \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \]

   if 1e3 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 56.6%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 44.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -200:\\ \;\;\;\;z \cdot \frac{y \cdot z}{x}\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 1000:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y 0.0007936500793651) -200.0)
   (* z (/ (* y z) x))
   (if (<= (+ y 0.0007936500793651) 1000.0)
     (* (* (/ z x) 0.0007936500793651) z)
     (* y (/ (* z z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -200.0) {
		tmp = z * ((y * z) / x);
	} else if ((y + 0.0007936500793651) <= 1000.0) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = y * ((z * z) / 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 + 0.0007936500793651d0) <= (-200.0d0)) then
        tmp = z * ((y * z) / x)
    else if ((y + 0.0007936500793651d0) <= 1000.0d0) then
        tmp = ((z / x) * 0.0007936500793651d0) * z
    else
        tmp = y * ((z * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -200.0) {
		tmp = z * ((y * z) / x);
	} else if ((y + 0.0007936500793651) <= 1000.0) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = y * ((z * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y + 0.0007936500793651) <= -200.0:
		tmp = z * ((y * z) / x)
	elif (y + 0.0007936500793651) <= 1000.0:
		tmp = ((z / x) * 0.0007936500793651) * z
	else:
		tmp = y * ((z * z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -200.0)
		tmp = Float64(z * Float64(Float64(y * z) / x));
	elseif (Float64(y + 0.0007936500793651) <= 1000.0)
		tmp = Float64(Float64(Float64(z / x) * 0.0007936500793651) * z);
	else
		tmp = Float64(y * Float64(Float64(z * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -200.0)
		tmp = z * ((y * z) / x);
	elseif ((y + 0.0007936500793651) <= 1000.0)
		tmp = ((z / x) * 0.0007936500793651) * z;
	else
		tmp = y * ((z * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -200.0], N[(z * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 1000.0], N[(N[(N[(z / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision], N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -200

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 54.1%

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

   if -200 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 1e3

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

      \[\leadsto \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \]

   if 1e3 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 56.6%

      \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.35e+60)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ z x) (+ 0.0007936500793651 y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.35e+60) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * (0.0007936500793651 + y)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.35e+60)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * Float64(0.0007936500793651 + y)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.35e+60], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e60

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if 1.35e60 < x

   1. Initial program 84.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 33.6%

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

Alternative 16: 31.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{z \cdot z}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * N[(N[(z * z), $MachinePrecision] / x), $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. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 33.7

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

5. Applied rewrites 33.7%

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

7. Applied rewrites 35.6%

   \[\leadsto y \cdot \color{blue}{\frac{z \cdot z}{x}} \]
8. Add Preprocessing

Developer Target 1: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024324 
(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)))