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

Percentage Accurate: 94.1% → 99.4%

Time: 14.7s

Alternatives: 12

Speedup: 1.0×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right) + \frac{1}{\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{y}{x}, \frac{\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right)}{x}\right), \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e-48)
   (+
    (fma -0.5 (log x) 0.91893853320467)
    (/
     1.0
     (/
      x
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333))))
   (+
    0.91893853320467
    (fma
     z
     (fma z (/ y x) (/ (fma z 0.0007936500793651 -0.0027777777777778) x))
     (fma (log x) (+ x -0.5) (- (/ 0.083333333333333 x) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e-48) {
		tmp = fma(-0.5, log(x), 0.91893853320467) + (1.0 / (x / fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333)));
	} else {
		tmp = 0.91893853320467 + fma(z, fma(z, (y / x), (fma(z, 0.0007936500793651, -0.0027777777777778) / x)), fma(log(x), (x + -0.5), ((0.083333333333333 / x) - x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e-48)
		tmp = Float64(fma(-0.5, log(x), 0.91893853320467) + Float64(1.0 / Float64(x / fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333))));
	else
		tmp = Float64(0.91893853320467 + fma(z, fma(z, Float64(y / x), Float64(fma(z, 0.0007936500793651, -0.0027777777777778) / x)), fma(log(x), Float64(x + -0.5), Float64(Float64(0.083333333333333 / x) - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e-48], N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(1.0 / N[(x / N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.91893853320467 + N[(z * N[(z * N[(y / x), $MachinePrecision] + N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.90000000000000001e-48

   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. *-commutative N/A

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

      8. lower-fma.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 1.90000000000000001e-48 < x

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.7%

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

Alternative 2: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ t_1 := t\_0 + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot y}{x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_0 + \frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))
        (t_1
         (+
          t_0
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_1 -2e+44)
     (* z (/ (* z y) x))
     (if (<= t_1 5e+306)
       (+ t_0 (/ 1.0 (* x 12.000000000000048)))
       (* (+ (/ y x) (/ 0.0007936500793651 x)) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double t_1 = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_1 <= -2e+44) {
		tmp = z * ((z * y) / x);
	} else if (t_1 <= 5e+306) {
		tmp = t_0 + (1.0 / (x * 12.000000000000048));
	} else {
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	}
	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 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    t_1 = t_0 + ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x)
    if (t_1 <= (-2d+44)) then
        tmp = z * ((z * y) / x)
    else if (t_1 <= 5d+306) then
        tmp = t_0 + (1.0d0 / (x * 12.000000000000048d0))
    else
        tmp = ((y / x) + (0.0007936500793651d0 / x)) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double t_1 = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_1 <= -2e+44) {
		tmp = z * ((z * y) / x);
	} else if (t_1 <= 5e+306) {
		tmp = t_0 + (1.0 / (x * 12.000000000000048));
	} else {
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	t_1 = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x)
	tmp = 0
	if t_1 <= -2e+44:
		tmp = z * ((z * y) / x)
	elif t_1 <= 5e+306:
		tmp = t_0 + (1.0 / (x * 12.000000000000048))
	else:
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	t_1 = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_1 <= -2e+44)
		tmp = Float64(z * Float64(Float64(z * y) / x));
	elseif (t_1 <= 5e+306)
		tmp = Float64(t_0 + Float64(1.0 / Float64(x * 12.000000000000048)));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	t_1 = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	tmp = 0.0;
	if (t_1 <= -2e+44)
		tmp = z * ((z * y) / x);
	elseif (t_1 <= 5e+306)
		tmp = t_0 + (1.0 / (x * 12.000000000000048));
	else
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+44], N[(z * N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], N[(t$95$0 + N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -2.0000000000000002e44

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 94.5%

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

   if -2.0000000000000002e44 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.99999999999999993e306

   1. Initial program 99.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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 99.5

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

      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. *-commutative N/A

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

      8. lower-fma.f64 99.5

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 4.99999999999999993e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 83.6%

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

   3. 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. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot y}{x}\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -2e+44)
     (* z (/ (* z y) x))
     (if (<= t_0 5e+306)
       (+
        0.91893853320467
        (fma (log x) (+ x -0.5) (- (/ 0.083333333333333 x) x)))
       (* (+ (/ y x) (/ 0.0007936500793651 x)) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -2e+44) {
		tmp = z * ((z * y) / x);
	} else if (t_0 <= 5e+306) {
		tmp = 0.91893853320467 + fma(log(x), (x + -0.5), ((0.083333333333333 / x) - x));
	} else {
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -2e+44)
		tmp = Float64(z * Float64(Float64(z * y) / x));
	elseif (t_0 <= 5e+306)
		tmp = Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(Float64(0.083333333333333 / x) - x)));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+44], N[(z * N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -2.0000000000000002e44

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 94.5%

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

   if -2.0000000000000002e44 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.99999999999999993e306

   1. Initial program 99.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 88.4

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

   10. Applied rewrites 88.4%

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

   if 4.99999999999999993e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 83.6%

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

   3. 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. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot y}{x}\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left(0.91893853320467 + \left(x \cdot \log x - x\right)\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -2e+44)
     (* z (/ (* z y) x))
     (if (<= t_0 5e+306)
       (+ (+ 0.91893853320467 (- (* x (log x)) x)) (/ 0.083333333333333 x))
       (* (+ (/ y x) (/ 0.0007936500793651 x)) (* z z))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -2e+44) {
		tmp = z * ((z * y) / x);
	} else if (t_0 <= 5e+306) {
		tmp = (0.91893853320467 + ((x * log(x)) - x)) + (0.083333333333333 / x);
	} else {
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	}
	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 = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x)
    if (t_0 <= (-2d+44)) then
        tmp = z * ((z * y) / x)
    else if (t_0 <= 5d+306) then
        tmp = (0.91893853320467d0 + ((x * log(x)) - x)) + (0.083333333333333d0 / x)
    else
        tmp = ((y / x) + (0.0007936500793651d0 / x)) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -2e+44) {
		tmp = z * ((z * y) / x);
	} else if (t_0 <= 5e+306) {
		tmp = (0.91893853320467 + ((x * Math.log(x)) - x)) + (0.083333333333333 / x);
	} else {
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x)
	tmp = 0
	if t_0 <= -2e+44:
		tmp = z * ((z * y) / x)
	elif t_0 <= 5e+306:
		tmp = (0.91893853320467 + ((x * math.log(x)) - x)) + (0.083333333333333 / x)
	else:
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -2e+44)
		tmp = Float64(z * Float64(Float64(z * y) / x));
	elseif (t_0 <= 5e+306)
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(x * log(x)) - x)) + Float64(0.083333333333333 / x));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.0007936500793651 / x)) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	tmp = 0.0;
	if (t_0 <= -2e+44)
		tmp = z * ((z * y) / x);
	elseif (t_0 <= 5e+306)
		tmp = (0.91893853320467 + ((x * log(x)) - x)) + (0.083333333333333 / x);
	else
		tmp = ((y / x) + (0.0007936500793651 / x)) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+44], N[(z * N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(N[(0.91893853320467 + N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -2.0000000000000002e44

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 94.5%

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

   if -2.0000000000000002e44 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 4.99999999999999993e306

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 86.9

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

   8. Applied rewrites 86.9%

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

   if 4.99999999999999993e306 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 83.6%

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

   3. 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. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \frac{z \cdot y}{x}\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\left(0.91893853320467 + \left(x \cdot \log x - x\right)\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;x \leq 430:\\ \;\;\;\;t\_0 + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= x 430.0)
     (+
      t_0
      (/
       (+
        0.083333333333333
        (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
       x))
     (+ t_0 (* z (* z (/ (+ y 0.0007936500793651) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 430.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (x <= 430.0d0) then
        tmp = t_0 + ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x)
    else
        tmp = t_0 + (z * (z * ((y + 0.0007936500793651d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 430.0) {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if x <= 430.0:
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x)
	else:
		tmp = t_0 + (z * (z * ((y + 0.0007936500793651) / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	tmp = 0.0
	if (x <= 430.0)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	tmp = 0.0;
	if (x <= 430.0)
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	else
		tmp = t_0 + (z * (z * ((y + 0.0007936500793651) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 430.0], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 430

   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 430 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 430:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \mathsf{fma}\left(x, 0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, -x\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e+15)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     (fma
      x
      (+ 0.91893853320467 (fma (log x) (+ x -0.5) (- x)))
      0.083333333333333))
    x)
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+15) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), fma(x, (0.91893853320467 + fma(log(x), (x + -0.5), -x)), 0.083333333333333)) / x;
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e+15)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), fma(x, Float64(0.91893853320467 + fma(log(x), Float64(x + -0.5), Float64(-x))), 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e+15], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + N[(x * N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.7%

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

   if 3e15 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), \mathsf{fma}\left(x, 0.91893853320467 + \mathsf{fma}\left(\log x, x + -0.5, -x\right), 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \mathsf{fma}\left(\log x, -0.5, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-7)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x)
    (fma (log x) -0.5 0.91893853320467))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (* z (/ (+ y 0.0007936500793651) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-7) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + fma(log(x), -0.5, 0.91893853320467);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y + 0.0007936500793651) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-7)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x) + fma(log(x), -0.5, 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e-7], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * -0.5 + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-7

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.9999999999999999e-7 < x

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \mathsf{fma}\left(\log x, -0.5, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8e+38)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+38) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(x, log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8e+38)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(x, log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.8e+38], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8e38

   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. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if 2.8e38 < x

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \log x + x \cdot -1} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \log x + \color{blue}{-1 \cdot x} \]

      8. neg-mul-1 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right)} \]

      10. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) \]

      11. lower-neg.f64 75.0

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

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, -x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in 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. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 61.7

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

5. Applied rewrites 61.7%

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

6. Final simplification 61.7%

   \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
7. Add Preprocessing

Alternative 10: 33.1% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 25.9

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

5. Applied rewrites 25.9%

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

7. Applied rewrites 27.0%

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

9. Applied rewrites 29.1%

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

10. Final simplification 29.1%

    \[\leadsto y \cdot \frac{z \cdot z}{x} \]
11. Add Preprocessing

Alternative 11: 30.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{y}{x} \cdot \left(z \cdot z\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 25.9

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

5. Applied rewrites 25.9%

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

7. Applied rewrites 28.1%

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

8. Final simplification 28.1%

   \[\leadsto \frac{y}{x} \cdot \left(z \cdot z\right) \]
9. Add Preprocessing

Alternative 12: 31.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot \frac{z}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 25.9

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

5. Applied rewrites 25.9%

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

7. Applied rewrites 27.0%

   \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{\frac{z}{x}} \]
8. 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 2024228 
(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)))