Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B

Percentage Accurate: 69.0% → 99.7%

Time: 11.2s

Alternatives: 15

Speedup: 4.2×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

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 + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

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 + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      2e+307)
   (+
    x
    (*
     y
     (/
      (fma (fma z 0.0692910599291889 0.4917317610505968) z 0.279195317918525)
      (fma (+ z 6.012459259764103) z 3.350343815022304))))
   (+ x (* y 0.0692910599291889))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= 2e+307) {
		tmp = x + (y * (fma(fma(z, 0.0692910599291889, 0.4917317610505968), z, 0.279195317918525) / fma((z + 6.012459259764103), z, 3.350343815022304)));
	} else {
		tmp = x + (y * 0.0692910599291889);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= 2e+307)
		tmp = Float64(x + Float64(y * Float64(fma(fma(z, 0.0692910599291889, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(z + 6.012459259764103), z, 3.350343815022304))));
	else
		tmp = Float64(x + Float64(y * 0.0692910599291889));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], 2e+307], N[(x + N[(y * N[(N[(N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(z + 6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.99999999999999997e307

   1. Initial program 97.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 97.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   if 1.99999999999999997e307 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 0.9%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 16.8%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 16.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 16.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

      2. *-commutative 99.6%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304\\ \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{t\_0} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\frac{t\_0}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))
   (if (<=
        (/
         (*
          y
          (+
           (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
           0.279195317918525))
         t_0)
        2e+307)
     (+
      x
      (/
       (fma z (fma z 0.0692910599291889 0.4917317610505968) 0.279195317918525)
       (/ t_0 y)))
     (+ x (* y 0.0692910599291889)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * (z + 6.012459259764103)) + 3.350343815022304;
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307) {
		tmp = x + (fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / (t_0 / y));
	} else {
		tmp = x + (y * 0.0692910599291889);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307)
		tmp = Float64(x + Float64(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / Float64(t_0 / y)));
	else
		tmp = Float64(x + Float64(y * 0.0692910599291889));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]}, If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 2e+307], N[(x + N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.99999999999999997e307

   1. Initial program 97.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 97.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      2. associate-*r/ 97.7%

         \[\leadsto x + \color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      3. fma-define 97.7%

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\color{blue}{z \cdot 0.0692910599291889 + 0.4917317610505968}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      4. fma-define 97.7%

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      5. *-commutative 97.7%

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      6. associate-*r/ 99.1%

         \[\leadsto x + \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      7. *-commutative 99.1%

         \[\leadsto x + \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304} \]

      8. fma-undefine 99.1%

         \[\leadsto x + \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}} \]

      9. clear-num 99.0%

         \[\leadsto x + \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{y}}} \]

      10. un-div-inv 99.1%

          \[\leadsto x + \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{y}}} \]

      11. *-commutative 99.1%

          \[\leadsto x + \frac{\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{y}} \]

      12. fma-undefine 99.1%

          \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{y}} \]

      13. fma-define 99.1%

          \[\leadsto x + \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right)}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{y}} \]

   6. Applied egg-rr 99.1%

      \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{y}}} \]

   7. Taylor expanded in y around 0 99.1%

      \[\leadsto x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\color{blue}{\frac{3.350343815022304 + z \cdot \left(6.012459259764103 + z\right)}{y}}} \]

   if 1.99999999999999997e307 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 0.9%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 16.8%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 16.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 16.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

      2. *-commutative 99.6%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\frac{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304\\ \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{t\_0} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y \cdot 0.279195317918525 + z \cdot \left(0.0692910599291889 \cdot \left(y \cdot z\right) + y \cdot 0.4917317610505968\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))
   (if (<=
        (/
         (*
          y
          (+
           (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
           0.279195317918525))
         t_0)
        2e+307)
     (+
      x
      (/
       (+
        (* y 0.279195317918525)
        (* z (+ (* 0.0692910599291889 (* y z)) (* y 0.4917317610505968))))
       t_0))
     (+ x (* y 0.0692910599291889)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * (z + 6.012459259764103)) + 3.350343815022304;
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307) {
		tmp = x + (((y * 0.279195317918525) + (z * ((0.0692910599291889 * (y * z)) + (y * 0.4917317610505968)))) / t_0);
	} else {
		tmp = x + (y * 0.0692910599291889);
	}
	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 = (z * (z + 6.012459259764103d0)) + 3.350343815022304d0
    if (((y * ((z * ((z * 0.0692910599291889d0) + 0.4917317610505968d0)) + 0.279195317918525d0)) / t_0) <= 2d+307) then
        tmp = x + (((y * 0.279195317918525d0) + (z * ((0.0692910599291889d0 * (y * z)) + (y * 0.4917317610505968d0)))) / t_0)
    else
        tmp = x + (y * 0.0692910599291889d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * (z + 6.012459259764103)) + 3.350343815022304;
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307) {
		tmp = x + (((y * 0.279195317918525) + (z * ((0.0692910599291889 * (y * z)) + (y * 0.4917317610505968)))) / t_0);
	} else {
		tmp = x + (y * 0.0692910599291889);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * (z + 6.012459259764103)) + 3.350343815022304
	tmp = 0
	if ((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307:
		tmp = x + (((y * 0.279195317918525) + (z * ((0.0692910599291889 * (y * z)) + (y * 0.4917317610505968)))) / t_0)
	else:
		tmp = x + (y * 0.0692910599291889)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307)
		tmp = Float64(x + Float64(Float64(Float64(y * 0.279195317918525) + Float64(z * Float64(Float64(0.0692910599291889 * Float64(y * z)) + Float64(y * 0.4917317610505968)))) / t_0));
	else
		tmp = Float64(x + Float64(y * 0.0692910599291889));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * (z + 6.012459259764103)) + 3.350343815022304;
	tmp = 0.0;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / t_0) <= 2e+307)
		tmp = x + (((y * 0.279195317918525) + (z * ((0.0692910599291889 * (y * z)) + (y * 0.4917317610505968)))) / t_0);
	else
		tmp = x + (y * 0.0692910599291889);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]}, If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 2e+307], N[(x + N[(N[(N[(y * 0.279195317918525), $MachinePrecision] + N[(z * N[(N[(0.0692910599291889 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(y * 0.4917317610505968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.99999999999999997e307

   1. Initial program 97.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 97.7%

      \[\leadsto x + \frac{\color{blue}{0.279195317918525 \cdot y + z \cdot \left(0.0692910599291889 \cdot \left(y \cdot z\right) + 0.4917317610505968 \cdot y\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   if 1.99999999999999997e307 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 0.9%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 16.8%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 16.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 16.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

      2. *-commutative 99.6%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y \cdot 0.279195317918525 + z \cdot \left(0.0692910599291889 \cdot \left(y \cdot z\right) + y \cdot 0.4917317610505968\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_0 + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
            0.279195317918525))
          (+ (* z (+ z 6.012459259764103)) 3.350343815022304))))
   (if (<= t_0 2e+307) (+ t_0 x) (+ x (* y 0.0692910599291889)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = t_0 + x;
	} else {
		tmp = x + (y * 0.0692910599291889);
	}
	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 = (y * ((z * ((z * 0.0692910599291889d0) + 0.4917317610505968d0)) + 0.279195317918525d0)) / ((z * (z + 6.012459259764103d0)) + 3.350343815022304d0)
    if (t_0 <= 2d+307) then
        tmp = t_0 + x
    else
        tmp = x + (y * 0.0692910599291889d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = t_0 + x;
	} else {
		tmp = x + (y * 0.0692910599291889);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)
	tmp = 0
	if t_0 <= 2e+307:
		tmp = t_0 + x
	else:
		tmp = x + (y * 0.0692910599291889)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= 2e+307)
		tmp = Float64(t_0 + x);
	else
		tmp = Float64(x + Float64(y * 0.0692910599291889));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	tmp = 0.0;
	if (t_0 <= 2e+307)
		tmp = t_0 + x;
	else
		tmp = x + (y * 0.0692910599291889);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+307], N[(t$95$0 + x), $MachinePrecision], N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.99999999999999997e307

   1. Initial program 97.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   if 1.99999999999999997e307 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 0.9%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 16.8%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 16.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 16.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 16.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

      2. *-commutative 99.6%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 - \frac{y \cdot -0.07512208616047561 + 0.4046220386999212 \cdot \frac{y}{z}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot \left(0.0007936505811533442 + z \cdot -0.0005951669793454025\right) - 0.00277777777751721\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -29000.0) (not (<= z 5.1e-8)))
   (+
    x
    (-
     (* y 0.0692910599291889)
     (/ (+ (* y -0.07512208616047561) (* 0.4046220386999212 (/ y z))) z)))
   (+
    x
    (*
     y
     (+
      0.08333333333333323
      (*
       z
       (-
        (* z (+ 0.0007936505811533442 (* z -0.0005951669793454025)))
        0.00277777777751721)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + ((y * 0.0692910599291889) - (((y * -0.07512208616047561) + (0.4046220386999212 * (y / z))) / z));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-29000.0d0)) .or. (.not. (z <= 5.1d-8))) then
        tmp = x + ((y * 0.0692910599291889d0) - (((y * (-0.07512208616047561d0)) + (0.4046220386999212d0 * (y / z))) / z))
    else
        tmp = x + (y * (0.08333333333333323d0 + (z * ((z * (0.0007936505811533442d0 + (z * (-0.0005951669793454025d0)))) - 0.00277777777751721d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + ((y * 0.0692910599291889) - (((y * -0.07512208616047561) + (0.4046220386999212 * (y / z))) / z));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -29000.0) or not (z <= 5.1e-8):
		tmp = x + ((y * 0.0692910599291889) - (((y * -0.07512208616047561) + (0.4046220386999212 * (y / z))) / z))
	else:
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -29000.0) || !(z <= 5.1e-8))
		tmp = Float64(x + Float64(Float64(y * 0.0692910599291889) - Float64(Float64(Float64(y * -0.07512208616047561) + Float64(0.4046220386999212 * Float64(y / z))) / z)));
	else
		tmp = Float64(x + Float64(y * Float64(0.08333333333333323 + Float64(z * Float64(Float64(z * Float64(0.0007936505811533442 + Float64(z * -0.0005951669793454025))) - 0.00277777777751721)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -29000.0) || ~((z <= 5.1e-8)))
		tmp = x + ((y * 0.0692910599291889) - (((y * -0.07512208616047561) + (0.4046220386999212 * (y / z))) / z));
	else
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -29000.0], N[Not[LessEqual[z, 5.1e-8]], $MachinePrecision]], N[(x + N[(N[(y * 0.0692910599291889), $MachinePrecision] - N[(N[(N[(y * -0.07512208616047561), $MachinePrecision] + N[(0.4046220386999212 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.08333333333333323 + N[(z * N[(N[(z * N[(0.0007936505811533442 + N[(z * -0.0005951669793454025), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -29000 or 5.10000000000000001e-8 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 41.5%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 53.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 53.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 53.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around -inf 98.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + -1 \cdot \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\left(-\frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)}\right) \]

      2. unsub-neg 98.6%

         \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 - \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)} \]

      3. sub-neg 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\color{blue}{0.4046220386999212 \cdot \frac{1}{z} + \left(-0.07512208616047561\right)}}{z}\right) \]

      4. associate-*r/ 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\color{blue}{\frac{0.4046220386999212 \cdot 1}{z}} + \left(-0.07512208616047561\right)}{z}\right) \]

      5. metadata-eval 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\frac{\color{blue}{0.4046220386999212}}{z} + \left(-0.07512208616047561\right)}{z}\right) \]

      6. metadata-eval 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} + \color{blue}{-0.07512208616047561}}{z}\right) \]

   7. Simplified 98.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} + -0.07512208616047561}{z}\right)} \]

   8. Taylor expanded in z around -inf 98.6%

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{-0.07512208616047561 \cdot y + 0.4046220386999212 \cdot \frac{y}{z}}{z} + 0.0692910599291889 \cdot y\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.7%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot \left(z \cdot \left(0.0007936505811533442 + -0.0005951669793454025 \cdot z\right) - 0.00277777777751721\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + \left(y \cdot 0.0692910599291889 - \frac{y \cdot -0.07512208616047561 + 0.4046220386999212 \cdot \frac{y}{z}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot \left(0.0007936505811533442 + z \cdot -0.0005951669793454025\right) - 0.00277777777751721\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 - \frac{-0.07512208616047561 + \frac{0.4046220386999212}{z}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot \left(0.0007936505811533442 + z \cdot -0.0005951669793454025\right) - 0.00277777777751721\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -29000.0) (not (<= z 5.1e-8)))
   (+
    x
    (*
     y
     (-
      0.0692910599291889
      (/ (+ -0.07512208616047561 (/ 0.4046220386999212 z)) z))))
   (+
    x
    (*
     y
     (+
      0.08333333333333323
      (*
       z
       (-
        (* z (+ 0.0007936505811533442 (* z -0.0005951669793454025)))
        0.00277777777751721)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-29000.0d0)) .or. (.not. (z <= 5.1d-8))) then
        tmp = x + (y * (0.0692910599291889d0 - (((-0.07512208616047561d0) + (0.4046220386999212d0 / z)) / z)))
    else
        tmp = x + (y * (0.08333333333333323d0 + (z * ((z * (0.0007936505811533442d0 + (z * (-0.0005951669793454025d0)))) - 0.00277777777751721d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -29000.0) or not (z <= 5.1e-8):
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)))
	else:
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -29000.0) || !(z <= 5.1e-8))
		tmp = Float64(x + Float64(y * Float64(0.0692910599291889 - Float64(Float64(-0.07512208616047561 + Float64(0.4046220386999212 / z)) / z))));
	else
		tmp = Float64(x + Float64(y * Float64(0.08333333333333323 + Float64(z * Float64(Float64(z * Float64(0.0007936505811533442 + Float64(z * -0.0005951669793454025))) - 0.00277777777751721)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -29000.0) || ~((z <= 5.1e-8)))
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)));
	else
		tmp = x + (y * (0.08333333333333323 + (z * ((z * (0.0007936505811533442 + (z * -0.0005951669793454025))) - 0.00277777777751721))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -29000.0], N[Not[LessEqual[z, 5.1e-8]], $MachinePrecision]], N[(x + N[(y * N[(0.0692910599291889 - N[(N[(-0.07512208616047561 + N[(0.4046220386999212 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.08333333333333323 + N[(z * N[(N[(z * N[(0.0007936505811533442 + N[(z * -0.0005951669793454025), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -29000 or 5.10000000000000001e-8 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 41.5%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 53.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 53.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 53.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around -inf 98.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + -1 \cdot \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\left(-\frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)}\right) \]

      2. unsub-neg 98.6%

         \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 - \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)} \]

      3. sub-neg 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\color{blue}{0.4046220386999212 \cdot \frac{1}{z} + \left(-0.07512208616047561\right)}}{z}\right) \]

      4. associate-*r/ 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\color{blue}{\frac{0.4046220386999212 \cdot 1}{z}} + \left(-0.07512208616047561\right)}{z}\right) \]

      5. metadata-eval 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\frac{\color{blue}{0.4046220386999212}}{z} + \left(-0.07512208616047561\right)}{z}\right) \]

      6. metadata-eval 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} + \color{blue}{-0.07512208616047561}}{z}\right) \]

   7. Simplified 98.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} + -0.07512208616047561}{z}\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.7%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot \left(z \cdot \left(0.0007936505811533442 + -0.0005951669793454025 \cdot z\right) - 0.00277777777751721\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 - \frac{-0.07512208616047561 + \frac{0.4046220386999212}{z}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot \left(0.0007936505811533442 + z \cdot -0.0005951669793454025\right) - 0.00277777777751721\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 - \frac{-0.07512208616047561 + \frac{0.4046220386999212}{z}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533442 - 0.00277777777751721\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -29000.0) (not (<= z 5.1e-8)))
   (+
    x
    (*
     y
     (-
      0.0692910599291889
      (/ (+ -0.07512208616047561 (/ 0.4046220386999212 z)) z))))
   (+
    x
    (*
     y
     (+
      0.08333333333333323
      (* z (- (* z 0.0007936505811533442) 0.00277777777751721)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-29000.0d0)) .or. (.not. (z <= 5.1d-8))) then
        tmp = x + (y * (0.0692910599291889d0 - (((-0.07512208616047561d0) + (0.4046220386999212d0 / z)) / z)))
    else
        tmp = x + (y * (0.08333333333333323d0 + (z * ((z * 0.0007936505811533442d0) - 0.00277777777751721d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -29000.0) or not (z <= 5.1e-8):
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)))
	else:
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -29000.0) || !(z <= 5.1e-8))
		tmp = Float64(x + Float64(y * Float64(0.0692910599291889 - Float64(Float64(-0.07512208616047561 + Float64(0.4046220386999212 / z)) / z))));
	else
		tmp = Float64(x + Float64(y * Float64(0.08333333333333323 + Float64(z * Float64(Float64(z * 0.0007936505811533442) - 0.00277777777751721)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -29000.0) || ~((z <= 5.1e-8)))
		tmp = x + (y * (0.0692910599291889 - ((-0.07512208616047561 + (0.4046220386999212 / z)) / z)));
	else
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -29000.0], N[Not[LessEqual[z, 5.1e-8]], $MachinePrecision]], N[(x + N[(y * N[(0.0692910599291889 - N[(N[(-0.07512208616047561 + N[(0.4046220386999212 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.08333333333333323 + N[(z * N[(N[(z * 0.0007936505811533442), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -29000 or 5.10000000000000001e-8 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 41.5%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 53.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 53.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 53.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around -inf 98.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + -1 \cdot \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\left(-\frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)}\right) \]

      2. unsub-neg 98.6%

         \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 - \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}\right)} \]

      3. sub-neg 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\color{blue}{0.4046220386999212 \cdot \frac{1}{z} + \left(-0.07512208616047561\right)}}{z}\right) \]

      4. associate-*r/ 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\color{blue}{\frac{0.4046220386999212 \cdot 1}{z}} + \left(-0.07512208616047561\right)}{z}\right) \]

      5. metadata-eval 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\frac{\color{blue}{0.4046220386999212}}{z} + \left(-0.07512208616047561\right)}{z}\right) \]

      6. metadata-eval 98.6%

         \[\leadsto x + y \cdot \left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} + \color{blue}{-0.07512208616047561}}{z}\right) \]

   7. Simplified 98.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} + -0.07512208616047561}{z}\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 - \frac{-0.07512208616047561 + \frac{0.4046220386999212}{z}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533442 - 0.00277777777751721\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000:\\ \;\;\;\;y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot -0.07512208616047561}{z}\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-8}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533442 - 0.00277777777751721\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000.0)
   (+ (* y 0.0692910599291889) (- x (/ (* y -0.07512208616047561) z)))
   (if (<= z 5.1e-8)
     (+
      x
      (*
       y
       (+
        0.08333333333333323
        (* z (- (* z 0.0007936505811533442) 0.00277777777751721)))))
     (+ x (* y (+ 0.0692910599291889 (/ 0.07512208616047561 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000.0) {
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z));
	} else if (z <= 5.1e-8) {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))));
	} else {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / 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) :: tmp
    if (z <= (-29000.0d0)) then
        tmp = (y * 0.0692910599291889d0) + (x - ((y * (-0.07512208616047561d0)) / z))
    else if (z <= 5.1d-8) then
        tmp = x + (y * (0.08333333333333323d0 + (z * ((z * 0.0007936505811533442d0) - 0.00277777777751721d0))))
    else
        tmp = x + (y * (0.0692910599291889d0 + (0.07512208616047561d0 / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000.0) {
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z));
	} else if (z <= 5.1e-8) {
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))));
	} else {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -29000.0:
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z))
	elif z <= 5.1e-8:
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))))
	else:
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000.0)
		tmp = Float64(Float64(y * 0.0692910599291889) + Float64(x - Float64(Float64(y * -0.07512208616047561) / z)));
	elseif (z <= 5.1e-8)
		tmp = Float64(x + Float64(y * Float64(0.08333333333333323 + Float64(z * Float64(Float64(z * 0.0007936505811533442) - 0.00277777777751721)))));
	else
		tmp = Float64(x + Float64(y * Float64(0.0692910599291889 + Float64(0.07512208616047561 / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -29000.0)
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z));
	elseif (z <= 5.1e-8)
		tmp = x + (y * (0.08333333333333323 + (z * ((z * 0.0007936505811533442) - 0.00277777777751721))));
	else
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000.0], N[(N[(y * 0.0692910599291889), $MachinePrecision] + N[(x - N[(N[(y * -0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e-8], N[(x + N[(y * N[(0.08333333333333323 + N[(z * N[(N[(z * 0.0007936505811533442), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.0692910599291889 + N[(0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -29000

   1. Initial program 47.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 47.5%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 47.5%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 58.6%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 58.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 58.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 58.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 58.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 58.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 58.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around -inf 99.2%

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z} + 0.0692910599291889 \cdot y\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 99.2%

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right) + 0.0692910599291889 \cdot y} \]

      2. +-commutative 99.2%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + \left(x + -1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right)} \]

      3. *-commutative 99.2%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + \left(x + -1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right) \]

      4. mul-1-neg 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \left(x + \color{blue}{\left(-\frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right)}\right) \]

      5. unsub-neg 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \color{blue}{\left(x - \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right)} \]

      6. distribute-rgt-out-- 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \left(x - \frac{\color{blue}{y \cdot \left(-0.4917317610505968 - -0.4166096748901212\right)}}{z}\right) \]

      7. metadata-eval 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot \color{blue}{-0.07512208616047561}}{z}\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot -0.07512208616047561}{z}\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.6%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right)\right)} \]

   if 5.10000000000000001e-8 < z

   1. Initial program 34.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 34.6%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 47.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 47.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 47.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 47.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 47.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 47.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 47.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 47.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 47.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 97.3%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.3%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\frac{0.07512208616047561 \cdot 1}{z}}\right) \]

      2. metadata-eval 97.3%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \frac{\color{blue}{0.07512208616047561}}{z}\right) \]

   7. Simplified 97.3%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000:\\ \;\;\;\;y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot -0.07512208616047561}{z}\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-8}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot \left(z \cdot 0.0007936505811533442 - 0.00277777777751721\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot -0.00277777777751721\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -29000.0) (not (<= z 5.1e-8)))
   (+ x (* y (+ 0.0692910599291889 (/ 0.07512208616047561 z))))
   (+ x (* y (+ 0.08333333333333323 (* z -0.00277777777751721))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-29000.0d0)) .or. (.not. (z <= 5.1d-8))) then
        tmp = x + (y * (0.0692910599291889d0 + (0.07512208616047561d0 / z)))
    else
        tmp = x + (y * (0.08333333333333323d0 + (z * (-0.00277777777751721d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	} else {
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -29000.0) or not (z <= 5.1e-8):
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)))
	else:
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -29000.0) || !(z <= 5.1e-8))
		tmp = Float64(x + Float64(y * Float64(0.0692910599291889 + Float64(0.07512208616047561 / z))));
	else
		tmp = Float64(x + Float64(y * Float64(0.08333333333333323 + Float64(z * -0.00277777777751721))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -29000.0) || ~((z <= 5.1e-8)))
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	else
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -29000.0], N[Not[LessEqual[z, 5.1e-8]], $MachinePrecision]], N[(x + N[(y * N[(0.0692910599291889 + N[(0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.08333333333333323 + N[(z * -0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -29000 or 5.10000000000000001e-8 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 41.5%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 53.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 53.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 53.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 98.4%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 98.4%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\frac{0.07512208616047561 \cdot 1}{z}}\right) \]

      2. metadata-eval 98.4%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \frac{\color{blue}{0.07512208616047561}}{z}\right) \]

   7. Simplified 98.4%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.5%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + -0.00277777777751721 \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto x + y \cdot \left(0.08333333333333323 + \color{blue}{z \cdot -0.00277777777751721}\right) \]

   7. Simplified 99.5%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot -0.00277777777751721\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot -0.00277777777751721\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.08333333333333323\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -29000.0) (not (<= z 5.1e-8)))
   (+ x (* y (+ 0.0692910599291889 (/ 0.07512208616047561 z))))
   (+ x (* y 0.08333333333333323))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	} else {
		tmp = x + (y * 0.08333333333333323);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-29000.0d0)) .or. (.not. (z <= 5.1d-8))) then
        tmp = x + (y * (0.0692910599291889d0 + (0.07512208616047561d0 / z)))
    else
        tmp = x + (y * 0.08333333333333323d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	} else {
		tmp = x + (y * 0.08333333333333323);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -29000.0) or not (z <= 5.1e-8):
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)))
	else:
		tmp = x + (y * 0.08333333333333323)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -29000.0) || !(z <= 5.1e-8))
		tmp = Float64(x + Float64(y * Float64(0.0692910599291889 + Float64(0.07512208616047561 / z))));
	else
		tmp = Float64(x + Float64(y * 0.08333333333333323));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -29000.0) || ~((z <= 5.1e-8)))
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	else
		tmp = x + (y * 0.08333333333333323);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -29000.0], N[Not[LessEqual[z, 5.1e-8]], $MachinePrecision]], N[(x + N[(y * N[(0.0692910599291889 + N[(0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.08333333333333323), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -29000 or 5.10000000000000001e-8 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 41.5%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 53.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 53.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 53.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 53.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 53.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 53.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 98.4%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 98.4%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\frac{0.07512208616047561 \cdot 1}{z}}\right) \]

      2. metadata-eval 98.4%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \frac{\color{blue}{0.07512208616047561}}{z}\right) \]

   7. Simplified 98.4%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 99.6%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 99.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 99.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.1%

      \[\leadsto \color{blue}{x + 0.08333333333333323 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{0.08333333333333323 \cdot y + x} \]

      2. *-commutative 99.1%

         \[\leadsto \color{blue}{y \cdot 0.08333333333333323} + x \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{y \cdot 0.08333333333333323 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.08333333333333323\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000:\\ \;\;\;\;y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot -0.07512208616047561}{z}\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-8}:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + z \cdot -0.00277777777751721\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000.0)
   (+ (* y 0.0692910599291889) (- x (/ (* y -0.07512208616047561) z)))
   (if (<= z 5.1e-8)
     (+ x (* y (+ 0.08333333333333323 (* z -0.00277777777751721))))
     (+ x (* y (+ 0.0692910599291889 (/ 0.07512208616047561 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000.0) {
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z));
	} else if (z <= 5.1e-8) {
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)));
	} else {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / 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) :: tmp
    if (z <= (-29000.0d0)) then
        tmp = (y * 0.0692910599291889d0) + (x - ((y * (-0.07512208616047561d0)) / z))
    else if (z <= 5.1d-8) then
        tmp = x + (y * (0.08333333333333323d0 + (z * (-0.00277777777751721d0))))
    else
        tmp = x + (y * (0.0692910599291889d0 + (0.07512208616047561d0 / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000.0) {
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z));
	} else if (z <= 5.1e-8) {
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)));
	} else {
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -29000.0:
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z))
	elif z <= 5.1e-8:
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)))
	else:
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000.0)
		tmp = Float64(Float64(y * 0.0692910599291889) + Float64(x - Float64(Float64(y * -0.07512208616047561) / z)));
	elseif (z <= 5.1e-8)
		tmp = Float64(x + Float64(y * Float64(0.08333333333333323 + Float64(z * -0.00277777777751721))));
	else
		tmp = Float64(x + Float64(y * Float64(0.0692910599291889 + Float64(0.07512208616047561 / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -29000.0)
		tmp = (y * 0.0692910599291889) + (x - ((y * -0.07512208616047561) / z));
	elseif (z <= 5.1e-8)
		tmp = x + (y * (0.08333333333333323 + (z * -0.00277777777751721)));
	else
		tmp = x + (y * (0.0692910599291889 + (0.07512208616047561 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000.0], N[(N[(y * 0.0692910599291889), $MachinePrecision] + N[(x - N[(N[(y * -0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e-8], N[(x + N[(y * N[(0.08333333333333323 + N[(z * -0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(0.0692910599291889 + N[(0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -29000

   1. Initial program 47.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 47.5%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 47.5%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 58.6%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 58.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 58.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 58.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 58.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 58.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 58.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around -inf 99.2%

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z} + 0.0692910599291889 \cdot y\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 99.2%

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right) + 0.0692910599291889 \cdot y} \]

      2. +-commutative 99.2%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + \left(x + -1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right)} \]

      3. *-commutative 99.2%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + \left(x + -1 \cdot \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right) \]

      4. mul-1-neg 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \left(x + \color{blue}{\left(-\frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right)}\right) \]

      5. unsub-neg 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \color{blue}{\left(x - \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}\right)} \]

      6. distribute-rgt-out-- 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \left(x - \frac{\color{blue}{y \cdot \left(-0.4917317610505968 - -0.4166096748901212\right)}}{z}\right) \]

      7. metadata-eval 99.2%

         \[\leadsto y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot \color{blue}{-0.07512208616047561}}{z}\right) \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + \left(x - \frac{y \cdot -0.07512208616047561}{z}\right)} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.7%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 99.8%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 99.8%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 99.8%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 99.8%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.5%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + -0.00277777777751721 \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto x + y \cdot \left(0.08333333333333323 + \color{blue}{z \cdot -0.00277777777751721}\right) \]

   7. Simplified 99.5%

      \[\leadsto x + y \cdot \color{blue}{\left(0.08333333333333323 + z \cdot -0.00277777777751721\right)} \]

   if 5.10000000000000001e-8 < z

   1. Initial program 34.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. remove-double-neg 34.6%

         \[\leadsto x + \color{blue}{\left(-\left(-\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      2. associate-/l* 47.3%

         \[\leadsto x + \left(-\left(-\color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}}\right)\right) \]

      3. distribute-rgt-neg-in 47.3%

         \[\leadsto x + \left(-\color{blue}{y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)}\right) \]

      4. distribute-lft-neg-in 47.3%

         \[\leadsto x + \color{blue}{\left(-y\right) \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)} \]

      5. distribute-lft-neg-in 47.3%

         \[\leadsto x + \color{blue}{\left(-y \cdot \left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      6. distribute-rgt-neg-in 47.3%

         \[\leadsto x + \color{blue}{y \cdot \left(-\left(-\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\right)\right)} \]

      7. remove-double-neg 47.3%

         \[\leadsto x + y \cdot \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]

      8. fma-define 47.3%

         \[\leadsto x + y \cdot \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      9. fma-define 47.3%

         \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      10. fma-define 47.3%

          \[\leadsto x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{x + y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 97.3%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.3%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \color{blue}{\frac{0.07512208616047561 \cdot 1}{z}}\right) \]

      2. metadata-eval 97.3%

         \[\leadsto x + y \cdot \left(0.0692910599291889 + \frac{\color{blue}{0.07512208616047561}}{z}\right) \]

   7. Simplified 97.3%

      \[\leadsto x + y \cdot \color{blue}{\left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.08333333333333323\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -29000.0) (not (<= z 5.1e-8)))
   (+ x (* y 0.0692910599291889))
   (+ x (* y 0.08333333333333323))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * 0.0692910599291889);
	} else {
		tmp = x + (y * 0.08333333333333323);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-29000.0d0)) .or. (.not. (z <= 5.1d-8))) then
        tmp = x + (y * 0.0692910599291889d0)
    else
        tmp = x + (y * 0.08333333333333323d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -29000.0) || !(z <= 5.1e-8)) {
		tmp = x + (y * 0.0692910599291889);
	} else {
		tmp = x + (y * 0.08333333333333323);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -29000.0) or not (z <= 5.1e-8):
		tmp = x + (y * 0.0692910599291889)
	else:
		tmp = x + (y * 0.08333333333333323)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -29000.0) || !(z <= 5.1e-8))
		tmp = Float64(x + Float64(y * 0.0692910599291889));
	else
		tmp = Float64(x + Float64(y * 0.08333333333333323));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -29000.0) || ~((z <= 5.1e-8)))
		tmp = x + (y * 0.0692910599291889);
	else
		tmp = x + (y * 0.08333333333333323);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -29000.0], N[Not[LessEqual[z, 5.1e-8]], $MachinePrecision]], N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.08333333333333323), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -29000 or 5.10000000000000001e-8 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 41.5%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 41.5%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 52.6%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 52.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 52.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 52.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 52.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 52.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 52.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 97.9%

      \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 97.9%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

      2. *-commutative 97.9%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

   7. Simplified 97.9%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]

   if -29000 < z < 5.10000000000000001e-8

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 99.6%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 99.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 99.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 99.6%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.1%

      \[\leadsto \color{blue}{x + 0.08333333333333323 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{0.08333333333333323 \cdot y + x} \]

      2. *-commutative 99.1%

         \[\leadsto \color{blue}{y \cdot 0.08333333333333323} + x \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{y \cdot 0.08333333333333323 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000 \lor \neg \left(z \leq 5.1 \cdot 10^{-8}\right):\\ \;\;\;\;x + y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot 0.08333333333333323\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-165}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4e+14) x (if (<= x 2.1e-165) (* y 0.0692910599291889) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e+14) {
		tmp = x;
	} else if (x <= 2.1e-165) {
		tmp = y * 0.0692910599291889;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4d+14)) then
        tmp = x
    else if (x <= 2.1d-165) then
        tmp = y * 0.0692910599291889d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e+14) {
		tmp = x;
	} else if (x <= 2.1e-165) {
		tmp = y * 0.0692910599291889;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4e+14:
		tmp = x
	elif x <= 2.1e-165:
		tmp = y * 0.0692910599291889
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4e+14)
		tmp = x;
	elseif (x <= 2.1e-165)
		tmp = Float64(y * 0.0692910599291889);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4e+14)
		tmp = x;
	elseif (x <= 2.1e-165)
		tmp = y * 0.0692910599291889;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4e+14], x, If[LessEqual[x, 2.1e-165], N[(y * 0.0692910599291889), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4e14 or 2.09999999999999995e-165 < x

   1. Initial program 72.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 72.5%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 72.5%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 76.0%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 76.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 76.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 76.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 76.0%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 76.0%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 76.0%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.0%

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

   if -4e14 < x < 2.09999999999999995e-165

   1. Initial program 68.1%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Step-by-step derivation

      1. +-commutative 68.1%

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

      2. *-commutative 68.1%

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

      3. associate-/l* 76.8%

         \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

      4. fma-define 76.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

      5. *-commutative 76.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      6. fma-define 76.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      7. fma-define 76.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

      8. *-commutative 76.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

      9. fma-define 76.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 67.6%

      \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative 67.6%

         \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

      2. *-commutative 67.6%

         \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

   7. Simplified 67.6%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 67.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot 0.0692910599291889} \cdot \sqrt[3]{y \cdot 0.0692910599291889}\right) \cdot \sqrt[3]{y \cdot 0.0692910599291889}} + x \]

      2. pow3 66.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot 0.0692910599291889}\right)}^{3}} + x \]

      3. *-commutative 66.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{0.0692910599291889 \cdot y}}\right)}^{3} + x \]

   9. Applied egg-rr 66.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{0.0692910599291889 \cdot y}\right)}^{3}} + x \]

   10. Taylor expanded in y around inf 66.8%

       \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} + {\left(\sqrt[3]{0.0692910599291889}\right)}^{3}\right)} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 67.6%

          \[\leadsto y \cdot \left(\frac{x}{y} + \color{blue}{0.0692910599291889}\right) \]

       2. +-commutative 67.6%

          \[\leadsto y \cdot \color{blue}{\left(0.0692910599291889 + \frac{x}{y}\right)} \]

   12. Simplified 67.6%

       \[\leadsto \color{blue}{y \cdot \left(0.0692910599291889 + \frac{x}{y}\right)} \]

   13. Taylor expanded in y around inf 52.2%

       \[\leadsto \color{blue}{0.0692910599291889 \cdot y} \]
   14. Step-by-step derivation

       1. *-commutative 52.2%

          \[\leadsto \color{blue}{y \cdot 0.0692910599291889} \]

   15. Simplified 52.2%

       \[\leadsto \color{blue}{y \cdot 0.0692910599291889} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ x + y \cdot 0.0692910599291889 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* y 0.0692910599291889)))

C

double code(double x, double y, double z) {
	return x + (y * 0.0692910599291889);
}

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 + (y * 0.0692910599291889d0)
end function

Java

public static double code(double x, double y, double z) {
	return x + (y * 0.0692910599291889);
}

Python

def code(x, y, z):
	return x + (y * 0.0692910599291889)

Julia

function code(x, y, z)
	return Float64(x + Float64(y * 0.0692910599291889))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y * 0.0692910599291889);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
2. Step-by-step derivation

   1. +-commutative 70.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

   2. *-commutative 70.8%

      \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

   3. associate-/l* 76.3%

      \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

   4. fma-define 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

   5. *-commutative 76.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

   6. fma-define 76.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

   7. fma-define 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

   8. *-commutative 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

   9. fma-define 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 79.8%

   \[\leadsto \color{blue}{x + 0.0692910599291889 \cdot y} \]
6. Step-by-step derivation

   1. +-commutative 79.8%

      \[\leadsto \color{blue}{0.0692910599291889 \cdot y + x} \]

   2. *-commutative 79.8%

      \[\leadsto \color{blue}{y \cdot 0.0692910599291889} + x \]

7. Simplified 79.8%

   \[\leadsto \color{blue}{y \cdot 0.0692910599291889 + x} \]

8. Final simplification 79.8%

   \[\leadsto x + y \cdot 0.0692910599291889 \]
9. Add Preprocessing

Alternative 15: 50.9% accurate, 21.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 70.8%

   \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
2. Step-by-step derivation

   1. +-commutative 70.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]

   2. *-commutative 70.8%

      \[\leadsto \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x \]

   3. associate-/l* 76.3%

      \[\leadsto \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]

   4. fma-define 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right)} \]

   5. *-commutative 76.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

   6. fma-define 76.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}, \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

   7. fma-define 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right), \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, x\right) \]

   8. *-commutative 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, x\right) \]

   9. fma-define 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, x\right) \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 51.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y)
          (- (/ (* 0.40462203869992125 y) (* z z)) x))))
   (if (< z -8120153.652456675)
     t_0
     (if (< z 6.576118972787377e+20)
       (+
        x
        (*
         (*
          y
          (+
           (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
           0.279195317918525))
         (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	double tmp;
	if (z < -8120153.652456675) {
		tmp = t_0;
	} else if (z < 6.576118972787377e+20) {
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((0.07512208616047561d0 / z) + 0.0692910599291889d0) * y) - (((0.40462203869992125d0 * y) / (z * z)) - x)
    if (z < (-8120153.652456675d0)) then
        tmp = t_0
    else if (z < 6.576118972787377d+20) then
        tmp = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) * (1.0d0 / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	double tmp;
	if (z < -8120153.652456675) {
		tmp = t_0;
	} else if (z < 6.576118972787377e+20) {
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x)
	tmp = 0
	if z < -8120153.652456675:
		tmp = t_0
	elif z < 6.576118972787377e+20:
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889) * y) - Float64(Float64(Float64(0.40462203869992125 * y) / Float64(z * z)) - x))
	tmp = 0.0
	if (z < -8120153.652456675)
		tmp = t_0;
	elseif (z < 6.576118972787377e+20)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * Float64(1.0 / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	tmp = 0.0;
	if (z < -8120153.652456675)
		tmp = t_0;
	elseif (z < 6.576118972787377e+20)
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(0.40462203869992125 * y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -8120153.652456675], t$95$0, If[Less[z, 6.576118972787377e+20], N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024111 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))