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

Percentage Accurate: 68.5% → 99.7%

Time: 3.7s

Alternatives: 10

Speedup: 2.1×

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 68.5% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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, 0.9× speedup

Math

\[\begin{array}{l} t_0 := x + \frac{1}{\frac{14.431876219268936}{y}}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 (/ 14.431876219268936 y)))))
   (if (<= z -2.3e+37)
     t_0
     (if (<= z 4e+18)
       (fma
        (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
        (/ y (fma (- z -6.012459259764103) z 3.350343815022304))
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (1.0 / (14.431876219268936 / y));
	double tmp;
	if (z <= -2.3e+37) {
		tmp = t_0;
	} else if (z <= 4e+18) {
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), (y / fma((z - -6.012459259764103), z, 3.350343815022304)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)))
	tmp = 0.0
	if (z <= -2.3e+37)
		tmp = t_0;
	elseif (z <= 4e+18)
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), Float64(y / fma(Float64(z - -6.012459259764103), z, 3.350343815022304)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.3e+37], t$95$0, If[LessEqual[z, 4e+18], N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * N[(y / N[(N[(z - -6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000002e37 or 4e18 < z

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. div-flip N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      4. lower-unsound-/.f64 68.5

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

      5. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      7. lower-fma.f64 68.5

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

      8. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      9. add-flip N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      11. metadata-eval 68.5

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

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}} \]

      14. lower-*.f64 68.5

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

      15. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      17. lower-fma.f64 68.5

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

      18. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      19. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      20. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      21. lower-fma.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{10000000000000000}{692910599291889}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.7

         \[\leadsto x + \frac{1}{\frac{14.431876219268936}{\color{blue}{y}}} \]

   6. Applied rewrites 79.7%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{14.431876219268936}{y}}} \]

   if -2.30000000000000002e37 < z < 4e18

   1. Initial program 68.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)\right) \cdot \frac{1}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)\right)} \cdot \frac{1}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y\right)} \cdot \frac{1}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot \left(y \cdot \frac{1}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}, y \cdot \frac{1}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, x\right)} \]

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, \frac{1}{\frac{1}{y}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{14.431876219268936}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      INFINITY)
   (fma
    (/
     (fma (fma z 0.0692910599291889 0.4917317610505968) z 0.279195317918525)
     (fma (- z -6.012459259764103) z 3.350343815022304))
    (/ 1.0 (/ 1.0 y))
    x)
   (+ x (/ 1.0 (/ 14.431876219268936 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= ((double) INFINITY)) {
		tmp = fma((fma(fma(z, 0.0692910599291889, 0.4917317610505968), z, 0.279195317918525) / fma((z - -6.012459259764103), z, 3.350343815022304)), (1.0 / (1.0 / y)), x);
	} else {
		tmp = x + (1.0 / (14.431876219268936 / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (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))) <= Inf)
		tmp = fma(Float64(fma(fma(z, 0.0692910599291889, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(z - -6.012459259764103), z, 3.350343815022304)), Float64(1.0 / Float64(1.0 / y)), x);
	else
		tmp = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[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], Infinity], N[(N[(N[(N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(z - -6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.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)))) < +inf.0

   1. Initial program 68.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right) + y \cdot \frac{11167812716741}{40000000000000}}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      7. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)} + x \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

   3. Applied rewrites 81.9%

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

   5. Applied rewrites 74.2%

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

   if +inf.0 < (+.f64 x (/.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 68.5%

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

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. div-flip N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      4. lower-unsound-/.f64 68.5

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

      5. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      7. lower-fma.f64 68.5

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

      8. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      9. add-flip N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      11. metadata-eval 68.5

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

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}} \]

      14. lower-*.f64 68.5

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

      15. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      17. lower-fma.f64 68.5

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

      18. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      19. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      20. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      21. lower-fma.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{10000000000000000}{692910599291889}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.7

         \[\leadsto x + \frac{1}{\frac{14.431876219268936}{\color{blue}{y}}} \]

   6. Applied rewrites 79.7%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{14.431876219268936}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;x + \mathsf{fma}\left(-1, \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}, 0.0692910599291889 \cdot y\right)\\ \mathbf{elif}\;z \leq 0.0122:\\ \;\;\;\;x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \left(0.14677053705526136 \cdot y - 0.14954831483277858 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{14.431876219268936}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -320000.0)
   (+
    x
    (fma
     -1.0
     (/ (- (* -0.4917317610505968 y) (* -0.4166096748901212 y)) z)
     (* 0.0692910599291889 y)))
   (if (<= z 0.0122)
     (+
      x
      (fma
       0.08333333333333323
       y
       (* z (- (* 0.14677053705526136 y) (* 0.14954831483277858 y)))))
     (+ x (/ 1.0 (/ 14.431876219268936 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -320000.0) {
		tmp = x + fma(-1.0, (((-0.4917317610505968 * y) - (-0.4166096748901212 * y)) / z), (0.0692910599291889 * y));
	} else if (z <= 0.0122) {
		tmp = x + fma(0.08333333333333323, y, (z * ((0.14677053705526136 * y) - (0.14954831483277858 * y))));
	} else {
		tmp = x + (1.0 / (14.431876219268936 / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -320000.0)
		tmp = Float64(x + fma(-1.0, Float64(Float64(Float64(-0.4917317610505968 * y) - Float64(-0.4166096748901212 * y)) / z), Float64(0.0692910599291889 * y)));
	elseif (z <= 0.0122)
		tmp = Float64(x + fma(0.08333333333333323, y, Float64(z * Float64(Float64(0.14677053705526136 * y) - Float64(0.14954831483277858 * y)))));
	else
		tmp = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -320000.0], N[(x + N[(-1.0 * N[(N[(N[(-0.4917317610505968 * y), $MachinePrecision] - N[(-0.4166096748901212 * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0122], N[(x + N[(0.08333333333333323 * y + N[(z * N[(N[(0.14677053705526136 * y), $MachinePrecision] - N[(0.14954831483277858 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2e5

   1. Initial program 68.5%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \color{blue}{\frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      2. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{z}}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      3. lower--.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      6. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

   if -3.2e5 < z < 0.0122000000000000008

   1. Initial program 68.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, \color{blue}{y}, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      5. lower-*.f64 65.3

         \[\leadsto x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \left(0.14677053705526136 \cdot y - 0.14954831483277858 \cdot y\right)\right) \]

   4. Applied rewrites 65.3%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, z \cdot \left(0.14677053705526136 \cdot y - 0.14954831483277858 \cdot y\right)\right)} \]

   if 0.0122000000000000008 < z

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. div-flip N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      4. lower-unsound-/.f64 68.5

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

      5. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      7. lower-fma.f64 68.5

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

      8. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      9. add-flip N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      11. metadata-eval 68.5

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

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}} \]

      14. lower-*.f64 68.5

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

      15. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      17. lower-fma.f64 68.5

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

      18. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      19. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      20. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      21. lower-fma.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{10000000000000000}{692910599291889}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.7

         \[\leadsto x + \frac{1}{\frac{14.431876219268936}{\color{blue}{y}}} \]

   6. Applied rewrites 79.7%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{14.431876219268936}{y}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := x + \frac{1}{\frac{14.431876219268936}{y}}\\ \mathbf{if}\;z \leq -320000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0122:\\ \;\;\;\;x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \left(0.14677053705526136 \cdot y - 0.14954831483277858 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 (/ 14.431876219268936 y)))))
   (if (<= z -320000.0)
     t_0
     (if (<= z 0.0122)
       (+
        x
        (fma
         0.08333333333333323
         y
         (* z (- (* 0.14677053705526136 y) (* 0.14954831483277858 y)))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (1.0 / (14.431876219268936 / y));
	double tmp;
	if (z <= -320000.0) {
		tmp = t_0;
	} else if (z <= 0.0122) {
		tmp = x + fma(0.08333333333333323, y, (z * ((0.14677053705526136 * y) - (0.14954831483277858 * y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)))
	tmp = 0.0
	if (z <= -320000.0)
		tmp = t_0;
	elseif (z <= 0.0122)
		tmp = Float64(x + fma(0.08333333333333323, y, Float64(z * Float64(Float64(0.14677053705526136 * y) - Float64(0.14954831483277858 * y)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -320000.0], t$95$0, If[LessEqual[z, 0.0122], N[(x + N[(0.08333333333333323 * y + N[(z * N[(N[(0.14677053705526136 * y), $MachinePrecision] - N[(0.14954831483277858 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e5 or 0.0122000000000000008 < z

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. div-flip N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      4. lower-unsound-/.f64 68.5

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

      5. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      7. lower-fma.f64 68.5

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

      8. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      9. add-flip N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      11. metadata-eval 68.5

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

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}} \]

      14. lower-*.f64 68.5

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

      15. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      17. lower-fma.f64 68.5

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

      18. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      19. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      20. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      21. lower-fma.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{10000000000000000}{692910599291889}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.7

         \[\leadsto x + \frac{1}{\frac{14.431876219268936}{\color{blue}{y}}} \]

   6. Applied rewrites 79.7%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{14.431876219268936}{y}}} \]

   if -3.2e5 < z < 0.0122000000000000008

   1. Initial program 68.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, \color{blue}{y}, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) \]

      5. lower-*.f64 65.3

         \[\leadsto x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \left(0.14677053705526136 \cdot y - 0.14954831483277858 \cdot y\right)\right) \]

   4. Applied rewrites 65.3%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, z \cdot \left(0.14677053705526136 \cdot y - 0.14954831483277858 \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := x + \frac{1}{\frac{14.431876219268936}{y}}\\ \mathbf{if}\;z \leq -63000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0125:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 (/ 14.431876219268936 y)))))
   (if (<= z -63000000.0)
     t_0
     (if (<= z 0.0125) (+ x (* 0.08333333333333323 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (1.0 / (14.431876219268936 / y));
	double tmp;
	if (z <= -63000000.0) {
		tmp = t_0;
	} else if (z <= 0.0125) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 / (14.431876219268936d0 / y))
    if (z <= (-63000000.0d0)) then
        tmp = t_0
    else if (z <= 0.0125d0) then
        tmp = x + (0.08333333333333323d0 * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (1.0 / (14.431876219268936 / y));
	double tmp;
	if (z <= -63000000.0) {
		tmp = t_0;
	} else if (z <= 0.0125) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (1.0 / (14.431876219268936 / y))
	tmp = 0
	if z <= -63000000.0:
		tmp = t_0
	elif z <= 0.0125:
		tmp = x + (0.08333333333333323 * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)))
	tmp = 0.0
	if (z <= -63000000.0)
		tmp = t_0;
	elseif (z <= 0.0125)
		tmp = Float64(x + Float64(0.08333333333333323 * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (1.0 / (14.431876219268936 / y));
	tmp = 0.0;
	if (z <= -63000000.0)
		tmp = t_0;
	elseif (z <= 0.0125)
		tmp = x + (0.08333333333333323 * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -63000000.0], t$95$0, If[LessEqual[z, 0.0125], N[(x + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.3e7 or 0.012500000000000001 < z

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. div-flip N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      4. lower-unsound-/.f64 68.5

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

      5. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      7. lower-fma.f64 68.5

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

      8. lift-+.f64 N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      9. add-flip N/A

         \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{6012459259764103}{1000000000000000}\right)\right)}, z, \frac{104698244219447}{31250000000000}\right)}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}} \]

      11. metadata-eval 68.5

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

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}} \]

      14. lower-*.f64 68.5

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

      15. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}} \]

      16. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      17. lower-fma.f64 68.5

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

      18. lift-+.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      19. lift-*.f64 N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      20. *-commutative N/A

          \[\leadsto x + \frac{1}{\frac{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}} \]

      21. lower-fma.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{10000000000000000}{692910599291889}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 79.7

         \[\leadsto x + \frac{1}{\frac{14.431876219268936}{\color{blue}{y}}} \]

   6. Applied rewrites 79.7%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{14.431876219268936}{y}}} \]

   if -6.3e7 < z < 0.012500000000000001

   1. Initial program 68.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 79.1

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

   4. Applied rewrites 79.1%

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

Alternative 6: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := x + 0.0692910599291889 \cdot y\\ \mathbf{if}\;z \leq -800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0125:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* 0.0692910599291889 y))))
   (if (<= z -800000000.0)
     t_0
     (if (<= z 0.0125) (+ x (* 0.08333333333333323 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (0.0692910599291889 * y);
	double tmp;
	if (z <= -800000000.0) {
		tmp = t_0;
	} else if (z <= 0.0125) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (0.0692910599291889d0 * y)
    if (z <= (-800000000.0d0)) then
        tmp = t_0
    else if (z <= 0.0125d0) then
        tmp = x + (0.08333333333333323d0 * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (0.0692910599291889 * y);
	double tmp;
	if (z <= -800000000.0) {
		tmp = t_0;
	} else if (z <= 0.0125) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (0.0692910599291889 * y)
	tmp = 0
	if z <= -800000000.0:
		tmp = t_0
	elif z <= 0.0125:
		tmp = x + (0.08333333333333323 * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(0.0692910599291889 * y))
	tmp = 0.0
	if (z <= -800000000.0)
		tmp = t_0;
	elseif (z <= 0.0125)
		tmp = Float64(x + Float64(0.08333333333333323 * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (0.0692910599291889 * y);
	tmp = 0.0;
	if (z <= -800000000.0)
		tmp = t_0;
	elseif (z <= 0.0125)
		tmp = x + (0.08333333333333323 * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -800000000.0], t$95$0, If[LessEqual[z, 0.0125], N[(x + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e8 or 0.012500000000000001 < z

   1. Initial program 68.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -8e8 < z < 0.012500000000000001

   1. Initial program 68.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 79.1

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

   4. Applied rewrites 79.1%

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

Alternative 7: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0692910599291889 \cdot z, z, 0.279195317918525\right)}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, \frac{1}{\frac{1}{y}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-1, \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}, 0.0692910599291889 \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      2e+304)
   (fma
    (/
     (fma (* 0.0692910599291889 z) z 0.279195317918525)
     (fma (- z -6.012459259764103) z 3.350343815022304))
    (/ 1.0 (/ 1.0 y))
    x)
   (+
    x
    (fma
     -1.0
     (/ (- (* -0.4917317610505968 y) (* -0.4166096748901212 y)) z)
     (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 2e+304) {
		tmp = fma((fma((0.0692910599291889 * z), z, 0.279195317918525) / fma((z - -6.012459259764103), z, 3.350343815022304)), (1.0 / (1.0 / y)), x);
	} else {
		tmp = x + fma(-1.0, (((-0.4917317610505968 * y) - (-0.4166096748901212 * y)) / z), (0.0692910599291889 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (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))) <= 2e+304)
		tmp = fma(Float64(fma(Float64(0.0692910599291889 * z), z, 0.279195317918525) / fma(Float64(z - -6.012459259764103), z, 3.350343815022304)), Float64(1.0 / Float64(1.0 / y)), x);
	else
		tmp = Float64(x + fma(-1.0, Float64(Float64(Float64(-0.4917317610505968 * y) - Float64(-0.4166096748901212 * y)) / z), Float64(0.0692910599291889 * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[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], 2e+304], N[(N[(N[(N[(0.0692910599291889 * z), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(z - -6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(-1.0 * N[(N[(N[(-0.4917317610505968 * y), $MachinePrecision] - N[(-0.4166096748901212 * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.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)))) < 1.9999999999999999e304

   1. Initial program 68.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right) + y \cdot \frac{11167812716741}{40000000000000}}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      7. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)} + x \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + \left(\frac{y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

   3. Applied rewrites 81.9%

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z}, z, \frac{11167812716741}{40000000000000}\right)}{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}, \frac{1}{\frac{1}{y}}, x\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 73.1

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

   8. Applied rewrites 73.1%

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

   if 1.9999999999999999e304 < (+.f64 x (/.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 68.5%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \color{blue}{\frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      2. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{z}}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      3. lower--.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(-1, \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}, \frac{692910599291889}{10000000000000000} \cdot y\right) \]

      6. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(-1, \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}, 0.0692910599291889 \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := x + 0.0692910599291889 \cdot y\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-117}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* 0.0692910599291889 y))))
   (if (<= z -1.55e-121)
     t_0
     (if (<= z 4.7e-117) (* y 0.08333333333333323) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (0.0692910599291889 * y);
	double tmp;
	if (z <= -1.55e-121) {
		tmp = t_0;
	} else if (z <= 4.7e-117) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (0.0692910599291889d0 * y)
    if (z <= (-1.55d-121)) then
        tmp = t_0
    else if (z <= 4.7d-117) then
        tmp = y * 0.08333333333333323d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (0.0692910599291889 * y);
	double tmp;
	if (z <= -1.55e-121) {
		tmp = t_0;
	} else if (z <= 4.7e-117) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (0.0692910599291889 * y)
	tmp = 0
	if z <= -1.55e-121:
		tmp = t_0
	elif z <= 4.7e-117:
		tmp = y * 0.08333333333333323
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(0.0692910599291889 * y))
	tmp = 0.0
	if (z <= -1.55e-121)
		tmp = t_0;
	elseif (z <= 4.7e-117)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (0.0692910599291889 * y);
	tmp = 0.0;
	if (z <= -1.55e-121)
		tmp = t_0;
	elseif (z <= 4.7e-117)
		tmp = y * 0.08333333333333323;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.55e-121], t$95$0, If[LessEqual[z, 4.7e-117], N[(y * 0.08333333333333323), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5499999999999999e-121 or 4.70000000000000009e-117 < z

   1. Initial program 68.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -1.5499999999999999e-121 < z < 4.70000000000000009e-117

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \color{blue}{\frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\color{blue}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + \color{blue}{z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \color{blue}{\left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + \color{blue}{z}\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      13. lower-+.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto y \cdot \frac{279195317918525}{3350343815022304} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.6%

      \[\leadsto y \cdot 0.08333333333333323 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -800000000:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq 0.0125:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -800000000.0)
   (* y 0.0692910599291889)
   (if (<= z 0.0125) (* y 0.08333333333333323) (* y 0.0692910599291889))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -800000000.0) {
		tmp = y * 0.0692910599291889;
	} else if (z <= 0.0125) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = y * 0.0692910599291889;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-800000000.0d0)) then
        tmp = y * 0.0692910599291889d0
    else if (z <= 0.0125d0) then
        tmp = y * 0.08333333333333323d0
    else
        tmp = y * 0.0692910599291889d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -800000000.0) {
		tmp = y * 0.0692910599291889;
	} else if (z <= 0.0125) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = y * 0.0692910599291889;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -800000000.0:
		tmp = y * 0.0692910599291889
	elif z <= 0.0125:
		tmp = y * 0.08333333333333323
	else:
		tmp = y * 0.0692910599291889
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -800000000.0)
		tmp = Float64(y * 0.0692910599291889);
	elseif (z <= 0.0125)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = Float64(y * 0.0692910599291889);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -800000000.0)
		tmp = y * 0.0692910599291889;
	elseif (z <= 0.0125)
		tmp = y * 0.08333333333333323;
	else
		tmp = y * 0.0692910599291889;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -800000000.0], N[(y * 0.0692910599291889), $MachinePrecision], If[LessEqual[z, 0.0125], N[(y * 0.08333333333333323), $MachinePrecision], N[(y * 0.0692910599291889), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e8 or 0.012500000000000001 < z

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \color{blue}{\frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\color{blue}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + \color{blue}{z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \color{blue}{\left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + \color{blue}{z}\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      13. lower-+.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in z around inf

      \[\leadsto y \cdot \frac{692910599291889}{10000000000000000} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.3%

      \[\leadsto y \cdot 0.0692910599291889 \]

   if -8e8 < z < 0.012500000000000001

   1. Initial program 68.5%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \color{blue}{\frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\color{blue}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + \color{blue}{z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \color{blue}{\left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + \color{blue}{z}\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

      13. lower-+.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto y \cdot \frac{279195317918525}{3350343815022304} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.6%

      \[\leadsto y \cdot 0.08333333333333323 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 31.3% accurate, 7.5× speedup

Math

\[y \cdot 0.0692910599291889 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * 0.0692910599291889d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

2. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto y \cdot \color{blue}{\left(\frac{11167812716741}{40000000000000} \cdot \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} + \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \color{blue}{\frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\color{blue}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + \color{blue}{z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \color{blue}{\left(\frac{6012459259764103}{1000000000000000} + z\right)}}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + \color{blue}{z}\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   9. lower-+.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto y \cdot \mathsf{fma}\left(\frac{11167812716741}{40000000000000}, \frac{1}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}, \frac{z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}\right) \]

   13. lower-+.f64 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in z around inf

   \[\leadsto y \cdot \frac{692910599291889}{10000000000000000} \]
6. Step-by-step derivation

7. Applied rewrites 31.3%

   \[\leadsto y \cdot 0.0692910599291889 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64
  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))