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

Percentage Accurate: 93.6% → 99.5%

Time: 12.9s

Alternatives: 19

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\frac{0.083333333333333 + \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z}{x} + \frac{{\left(\mathsf{fma}\left(\log x, x - 0.5, -x\right)\right)}^{2} - 0.8444480278083504}{\mathsf{fma}\left(\log x, x - 0.5, \left(-x\right) - 0.91893853320467\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e+29)
   (+
    (/
     (+
      0.083333333333333
      (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z))
     x)
    (/
     (- (pow (fma (log x) (- x 0.5) (- x)) 2.0) 0.8444480278083504)
     (fma (log x) (- x 0.5) (- (- x) 0.91893853320467))))
   (-
    (+
     (fma (- x 0.5) (log x) 0.91893853320467)
     (* (* (+ 0.0007936500793651 y) (/ z x)) z))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+29) {
		tmp = ((0.083333333333333 + ((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z)) / x) + ((pow(fma(log(x), (x - 0.5), -x), 2.0) - 0.8444480278083504) / fma(log(x), (x - 0.5), (-x - 0.91893853320467)));
	} else {
		tmp = (fma((x - 0.5), log(x), 0.91893853320467) + (((0.0007936500793651 + y) * (z / x)) * z)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e+29)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z)) / x) + Float64(Float64((fma(log(x), Float64(x - 0.5), Float64(-x)) ^ 2.0) - 0.8444480278083504) / fma(log(x), Float64(x - 0.5), Float64(Float64(-x) - 0.91893853320467))));
	else
		tmp = Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z / x)) * z)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e+29], N[(N[(N[(0.083333333333333 + N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[Power[N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + (-x)), $MachinePrecision], 2.0], $MachinePrecision] - 0.8444480278083504), $MachinePrecision] / N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[((-x) - 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9999999999999999e29

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, -x\right)\right)}^{2} - \color{blue}{\frac{8444480278083503881401098089}{10000000000000000000000000000}}}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) - \frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      14. lift--.f64 N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, -x\right)\right)}^{2} - \frac{8444480278083503881401098089}{10000000000000000000000000000}}{\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} - \frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      15. sub-neg N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(\log x, x - \frac{1}{2}, -x\right)\right)}^{2} - \frac{8444480278083503881401098089}{10000000000000000000000000000}}{\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} - \frac{91893853320467}{100000000000000}} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(\log x, x - 0.5, -x\right)\right)}^{2} - 0.8444480278083504}{\mathsf{fma}\left(\log x, x - 0.5, \left(-x\right) - 0.91893853320467\right)}} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 2.9999999999999999e29 < x

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\frac{0.083333333333333 + \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z}{x} + \frac{{\left(\mathsf{fma}\left(\log x, x - 0.5, -x\right)\right)}^{2} - 0.8444480278083504}{\mathsf{fma}\left(\log x, x - 0.5, \left(-x\right) - 0.91893853320467\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.0007936500793651 + y\right) \cdot z\\ t_1 := \left(t\_0 - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{1}{12.000000000000048 \cdot x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ 0.0007936500793651 y) z))
        (t_1 (* (- t_0 0.0027777777777778) z)))
   (if (<= t_1 -5e+14)
     (* (* (/ z x) z) (+ 0.0007936500793651 y))
     (if (<= t_1 4e+49)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 1.0 (* 12.000000000000048 x)) 0.91893853320467) x))
       (* t_0 (/ z x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) * z;
	double t_1 = (t_0 - 0.0027777777777778) * z;
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	} else if (t_1 <= 4e+49) {
		tmp = fma((x - 0.5), log(x), (((1.0 / (12.000000000000048 * x)) + 0.91893853320467) - x));
	} else {
		tmp = t_0 * (z / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) * z)
	t_1 = Float64(Float64(t_0 - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_1 <= -5e+14)
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	elseif (t_1 <= 4e+49)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(1.0 / Float64(12.000000000000048 * x)) + 0.91893853320467) - x));
	else
		tmp = Float64(t_0 * Float64(z / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+14], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+49], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(z / x), $MachinePrecision]), $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) < -5e14

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 86.3%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 3.99999999999999979e49

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\frac{1}{x \cdot 12.000000000000048} + 0.91893853320467\right) - x\right) \]

   if 3.99999999999999979e49 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{1}{12.000000000000048 \cdot x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot \frac{z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.0007936500793651 + y\right) \cdot z\\ t_1 := \left(t\_0 - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ 0.0007936500793651 y) z))
        (t_1 (* (- t_0 0.0027777777777778) z)))
   (if (<= t_1 -5e+14)
     (* (* (/ z x) z) (+ 0.0007936500793651 y))
     (if (<= t_1 4e+49)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (* t_0 (/ z x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) * z;
	double t_1 = (t_0 - 0.0027777777777778) * z;
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	} else if (t_1 <= 4e+49) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = t_0 * (z / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) * z)
	t_1 = Float64(Float64(t_0 - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_1 <= -5e+14)
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	elseif (t_1 <= 4e+49)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = Float64(t_0 * Float64(z / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+14], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+49], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(z / x), $MachinePrecision]), $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) < -5e14

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 86.3%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 3.99999999999999979e49

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 96.8

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

   5. Applied rewrites 96.8%

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

   if 3.99999999999999979e49 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot \frac{z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.0007936500793651 + y\right) \cdot z\\ t_1 := \left(t\_0 - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right) + \left(\frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ 0.0007936500793651 y) z))
        (t_1 (* (- t_0 0.0027777777777778) z)))
   (if (<= t_1 -5e+14)
     (* (* (/ z x) z) (+ 0.0007936500793651 y))
     (if (<= t_1 4e+49)
       (+
        (fma (log x) (- x 0.5) 0.91893853320467)
        (- (/ 0.083333333333333 x) x))
       (* t_0 (/ z x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) * z;
	double t_1 = (t_0 - 0.0027777777777778) * z;
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	} else if (t_1 <= 4e+49) {
		tmp = fma(log(x), (x - 0.5), 0.91893853320467) + ((0.083333333333333 / x) - x);
	} else {
		tmp = t_0 * (z / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) * z)
	t_1 = Float64(Float64(t_0 - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_1 <= -5e+14)
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	elseif (t_1 <= 4e+49)
		tmp = Float64(fma(log(x), Float64(x - 0.5), 0.91893853320467) + Float64(Float64(0.083333333333333 / x) - x));
	else
		tmp = Float64(t_0 * Float64(z / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+14], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+49], N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(z / x), $MachinePrecision]), $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) < -5e14

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 86.3%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 3.99999999999999979e49

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-+r+ N/A

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

      6. associate-+l+ N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 96.6%

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

   if 3.99999999999999979e49 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right) + \left(\frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot \frac{z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+28)
   (+
    (/
     1.0
     (/
      x
      (fma
       (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
       z
       0.083333333333333)))
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))
   (-
    (+
     (fma (- x 0.5) (log x) 0.91893853320467)
     (* (* (+ 0.0007936500793651 y) (/ z x)) z))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+28) {
		tmp = (1.0 / (x / fma(fma(z, (0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333))) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	} else {
		tmp = (fma((x - 0.5), log(x), 0.91893853320467) + (((0.0007936500793651 + y) * (z / x)) * z)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+28)
		tmp = Float64(Float64(1.0 / Float64(x / fma(fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333))) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	else
		tmp = Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z / x)) * z)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+28], N[(N[(1.0 / N[(x / N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999992e28

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.7

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.99999999999999992e28 < x

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+14)
   (+
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
    (/
     (+
      0.083333333333333
      (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z))
     x))
   (-
    (+
     (fma (- x 0.5) (log x) 0.91893853320467)
     (* (* (+ 0.0007936500793651 y) (/ z x)) z))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+14) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + ((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z)) / x);
	} else {
		tmp = (fma((x - 0.5), log(x), 0.91893853320467) + (((0.0007936500793651 + y) * (z / x)) * z)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+14)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z)) / x));
	else
		tmp = Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z / x)) * z)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+14], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 + N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e14

   1. Initial program 99.7%

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

   if 2e14 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.022:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.022)
   (+
    (fma -0.5 (log x) 0.91893853320467)
    (/
     1.0
     (/
      x
      (fma
       (fma z (+ 0.0007936500793651 y) -0.0027777777777778)
       z
       0.083333333333333))))
   (-
    (+
     (fma (- x 0.5) (log x) 0.91893853320467)
     (* (* (+ 0.0007936500793651 y) (/ z x)) z))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.022) {
		tmp = fma(-0.5, log(x), 0.91893853320467) + (1.0 / (x / fma(fma(z, (0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333)));
	} else {
		tmp = (fma((x - 0.5), log(x), 0.91893853320467) + (((0.0007936500793651 + y) * (z / x)) * z)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.022)
		tmp = Float64(fma(-0.5, log(x), 0.91893853320467) + Float64(1.0 / Float64(x / fma(fma(z, Float64(0.0007936500793651 + y), -0.0027777777777778), z, 0.083333333333333))));
	else
		tmp = Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z / x)) * z)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.022], N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(1.0 / N[(x / N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.021999999999999999

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.7

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 0.021999999999999999 < x

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.022:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.022:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.022)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (-
    (+
     (fma (- x 0.5) (log x) 0.91893853320467)
     (* (* (+ 0.0007936500793651 y) (/ z x)) z))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.022) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (fma((x - 0.5), log(x), 0.91893853320467) + (((0.0007936500793651 + y) * (z / x)) * z)) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.022)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(Float64(Float64(0.0007936500793651 + y) * Float64(z / x)) * z)) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.022], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.021999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 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 0.021999999999999999 < x

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.022:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\left(0.0007936500793651 + y\right) \cdot \frac{z}{x}\right) \cdot z\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.5e+60)
   (/
    (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 <= 4.5e+60) {
		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 <= 4.5e+60)
		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, 4.5e+60], 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 < 4.50000000000000013e60

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if 4.50000000000000013e60 < x

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   7. 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. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-log.f64 69.6

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

   8. Applied rewrites 69.6%

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

Alternative 10: 63.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.0007936500793651 + y\right) \cdot z\\ t_1 := \left(t\_0 - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ 0.0007936500793651 y) z))
        (t_1 (* (- t_0 0.0027777777777778) z)))
   (if (<= t_1 -5e+14)
     (* (* (/ z x) z) (+ 0.0007936500793651 y))
     (if (<= t_1 2e-6) (/ 1.0 (* 12.000000000000048 x)) (* t_0 (/ z x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) * z;
	double t_1 = (t_0 - 0.0027777777777778) * z;
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	} else if (t_1 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.0007936500793651d0 + y) * z
    t_1 = (t_0 - 0.0027777777777778d0) * z
    if (t_1 <= (-5d+14)) then
        tmp = ((z / x) * z) * (0.0007936500793651d0 + y)
    else if (t_1 <= 2d-6) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = t_0 * (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.0007936500793651 + y) * z;
	double t_1 = (t_0 - 0.0027777777777778) * z;
	double tmp;
	if (t_1 <= -5e+14) {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	} else if (t_1 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = t_0 * (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.0007936500793651 + y) * z
	t_1 = (t_0 - 0.0027777777777778) * z
	tmp = 0
	if t_1 <= -5e+14:
		tmp = ((z / x) * z) * (0.0007936500793651 + y)
	elif t_1 <= 2e-6:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = t_0 * (z / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0007936500793651 + y) * z)
	t_1 = Float64(Float64(t_0 - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_1 <= -5e+14)
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	elseif (t_1 <= 2e-6)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(t_0 * Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.0007936500793651 + y) * z;
	t_1 = (t_0 - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_1 <= -5e+14)
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	elseif (t_1 <= 2e-6)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = t_0 * (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+14], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-6], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(z / x), $MachinePrecision]), $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) < -5e14

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 86.3%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.99999999999999991e-6

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.4%

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

   10. Applied rewrites 42.4%

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

   if 1.99999999999999991e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 79.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot \frac{z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z))
        (t_1 (* (* (/ z x) z) (+ 0.0007936500793651 y))))
   (if (<= t_0 -5e+14)
     t_1
     (if (<= t_0 2e-6) (/ 1.0 (* 12.000000000000048 x)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double t_1 = ((z / x) * z) * (0.0007936500793651 + y);
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = t_1;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((0.0007936500793651d0 + y) * z) - 0.0027777777777778d0) * z
    t_1 = ((z / x) * z) * (0.0007936500793651d0 + y)
    if (t_0 <= (-5d+14)) then
        tmp = t_1
    else if (t_0 <= 2d-6) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double t_1 = ((z / x) * z) * (0.0007936500793651 + y);
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = t_1;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z
	t_1 = ((z / x) * z) * (0.0007936500793651 + y)
	tmp = 0
	if t_0 <= -5e+14:
		tmp = t_1
	elif t_0 <= 2e-6:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y))
	tmp = 0.0
	if (t_0 <= -5e+14)
		tmp = t_1;
	elseif (t_0 <= 2e-6)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	t_1 = ((z / x) * z) * (0.0007936500793651 + y);
	tmp = 0.0;
	if (t_0 <= -5e+14)
		tmp = t_1;
	elseif (t_0 <= 2e-6)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+14], t$95$1, If[LessEqual[t$95$0, 2e-6], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -5e14 or 1.99999999999999991e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.8%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 80.7%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.99999999999999991e-6

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.4%

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

   10. Applied rewrites 42.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \frac{z}{x} \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_1 \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z))
        (t_1 (* (/ z x) z)))
   (if (<= t_0 -5e+14)
     (* t_1 y)
     (if (<= t_0 2e-6)
       (/ 1.0 (* 12.000000000000048 x))
       (* t_1 0.0007936500793651)))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double t_1 = (z / x) * z;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = t_1 * y;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = t_1 * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((0.0007936500793651d0 + y) * z) - 0.0027777777777778d0) * z
    t_1 = (z / x) * z
    if (t_0 <= (-5d+14)) then
        tmp = t_1 * y
    else if (t_0 <= 2d-6) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = t_1 * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double t_1 = (z / x) * z;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = t_1 * y;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = t_1 * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z
	t_1 = (z / x) * z
	tmp = 0
	if t_0 <= -5e+14:
		tmp = t_1 * y
	elif t_0 <= 2e-6:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = t_1 * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(z / x) * z)
	tmp = 0.0
	if (t_0 <= -5e+14)
		tmp = Float64(t_1 * y);
	elseif (t_0 <= 2e-6)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(t_1 * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	t_1 = (z / x) * z;
	tmp = 0.0;
	if (t_0 <= -5e+14)
		tmp = t_1 * y;
	elseif (t_0 <= 2e-6)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = t_1 * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+14], N[(t$95$1 * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 0.0007936500793651), $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) < -5e14

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.1%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.99999999999999991e-6

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.4%

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

   10. Applied rewrites 42.4%

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

   if 1.99999999999999991e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.3%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{\color{blue}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z)))
   (if (<= t_0 -5e+14)
     (* (* (/ z x) y) z)
     (if (<= t_0 2e-6)
       (/ 1.0 (* 12.000000000000048 x))
       (* (* (/ z x) z) 0.0007936500793651)))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((0.0007936500793651d0 + y) * z) - 0.0027777777777778d0) * z
    if (t_0 <= (-5d+14)) then
        tmp = ((z / x) * y) * z
    else if (t_0 <= 2d-6) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z
	tmp = 0
	if t_0 <= -5e+14:
		tmp = ((z / x) * y) * z
	elif t_0 <= 2e-6:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -5e+14)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 2e-6)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_0 <= -5e+14)
		tmp = ((z / x) * y) * z;
	elseif (t_0 <= 2e-6)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+14], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $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) < -5e14

   1. Initial program 87.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.6%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.99999999999999991e-6

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.4%

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

   10. Applied rewrites 42.4%

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

   if 1.99999999999999991e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.3%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{\color{blue}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z)))
   (if (<= t_0 -5e+14)
     (* (* (/ y x) z) z)
     (if (<= t_0 2e-6)
       (/ 1.0 (* 12.000000000000048 x))
       (* (* (/ z x) z) 0.0007936500793651)))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = ((y / x) * z) * z;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((0.0007936500793651d0 + y) * z) - 0.0027777777777778d0) * z
    if (t_0 <= (-5d+14)) then
        tmp = ((y / x) * z) * z
    else if (t_0 <= 2d-6) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -5e+14) {
		tmp = ((y / x) * z) * z;
	} else if (t_0 <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z
	tmp = 0
	if t_0 <= -5e+14:
		tmp = ((y / x) * z) * z
	elif t_0 <= 2e-6:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -5e+14)
		tmp = Float64(Float64(Float64(y / x) * z) * z);
	elseif (t_0 <= 2e-6)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.0007936500793651 + y) * z) - 0.0027777777777778) * z;
	tmp = 0.0;
	if (t_0 <= -5e+14)
		tmp = ((y / x) * z) * z;
	elseif (t_0 <= 2e-6)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+14], N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $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) < -5e14

   1. Initial program 87.2%

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

   3. Taylor expanded in 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 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 77.8%

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

   if -5e14 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.99999999999999991e-6

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.4%

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

   10. Applied rewrites 42.4%

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

   if 1.99999999999999991e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.3%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{\color{blue}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \mathbf{elif}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ 0.0007936500793651 y) z) 0.0027777777777778) z) 2e-6)
   (/ 1.0 (* 12.000000000000048 x))
   (* (* (/ z x) z) 0.0007936500793651)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z) <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((0.0007936500793651d0 + y) * z) - 0.0027777777777778d0) * z) <= 2d-6) then
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    else
        tmp = ((z / x) * z) * 0.0007936500793651d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z) <= 2e-6) {
		tmp = 1.0 / (12.000000000000048 * x);
	} else {
		tmp = ((z / x) * z) * 0.0007936500793651;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z) <= 2e-6:
		tmp = 1.0 / (12.000000000000048 * x)
	else:
		tmp = ((z / x) * z) * 0.0007936500793651
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) - 0.0027777777777778) * z) <= 2e-6)
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((((0.0007936500793651 + y) * z) - 0.0027777777777778) * z) <= 2e-6)
		tmp = 1.0 / (12.000000000000048 * x);
	else
		tmp = ((z / x) * z) * 0.0007936500793651;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 2e-6], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.99999999999999991e-6

   1. Initial program 96.1%

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

   3. 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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.5%

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

   10. Applied rewrites 31.5%

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

   if 1.99999999999999991e-6 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.3%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{\color{blue}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(0.0007936500793651 + y\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.0007936500793651 + y\right) \cdot z}{x} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.3e+61)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (/ (* (+ 0.0007936500793651 y) z) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.3e+61) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((0.0007936500793651 + y) * z) / x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.3e+61)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 + y) * z) / x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.3e+61], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3e61

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 2.3e61 < x

   1. Initial program 83.9%

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

   3. 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 32.0

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

   5. Applied rewrites 32.0%

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

   7. Applied rewrites 32.2%

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

Alternative 17: 26.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -30:\\ \;\;\;\;\frac{z}{x} \cdot -0.0027777777777778\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -30.0)
   (* (/ z x) -0.0027777777777778)
   (/ 1.0 (* 12.000000000000048 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -30.0) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 1.0 / (12.000000000000048 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-30.0d0)) then
        tmp = (z / x) * (-0.0027777777777778d0)
    else
        tmp = 1.0d0 / (12.000000000000048d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -30.0) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 1.0 / (12.000000000000048 * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -30.0:
		tmp = (z / x) * -0.0027777777777778
	else:
		tmp = 1.0 / (12.000000000000048 * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -30.0)
		tmp = Float64(Float64(z / x) * -0.0027777777777778);
	else
		tmp = Float64(1.0 / Float64(12.000000000000048 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -30.0)
		tmp = (z / x) * -0.0027777777777778;
	else
		tmp = 1.0 / (12.000000000000048 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -30.0], N[(N[(z / x), $MachinePrecision] * -0.0027777777777778), $MachinePrecision], N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -30

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 24.0%

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

   if -30 < z

   1. Initial program 95.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.1%

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

   10. Applied rewrites 25.1%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -30:\\ \;\;\;\;\frac{z}{x} \cdot -0.0027777777777778\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12.000000000000048 \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -30:\\ \;\;\;\;\frac{z}{x} \cdot -0.0027777777777778\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -30.0) (* (/ z x) -0.0027777777777778) (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -30.0) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-30.0d0)) then
        tmp = (z / x) * (-0.0027777777777778d0)
    else
        tmp = 0.083333333333333d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -30.0) {
		tmp = (z / x) * -0.0027777777777778;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -30.0:
		tmp = (z / x) * -0.0027777777777778
	else:
		tmp = 0.083333333333333 / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -30.0)
		tmp = Float64(Float64(z / x) * -0.0027777777777778);
	else
		tmp = Float64(0.083333333333333 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -30.0)
		tmp = (z / x) * -0.0027777777777778;
	else
		tmp = 0.083333333333333 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -30.0], N[(N[(z / x), $MachinePrecision] * -0.0027777777777778), $MachinePrecision], N[(0.083333333333333 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -30

   1. Initial program 87.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 24.0%

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

   if -30 < z

   1. Initial program 95.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.1%

      \[\leadsto \frac{0.083333333333333}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 22.4% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 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 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 19.5%

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

Developer Target 1: 98.7% 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 2024294 
(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)))