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

Percentage Accurate: 68.6% → 99.7%

Time: 6.0s

Alternatives: 10

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(0.0692910599291889, y, -\frac{\mathsf{fma}\left(\frac{0.279195317918525 \cdot y - \mathsf{fma}\left(-6.012459259764103, y \cdot -0.07512208616047561, 0.23214887407009785 \cdot y\right)}{z}, -1, y \cdot -0.07512208616047561\right)}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      5e+307)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (+ 6.012459259764103 z) z 3.350343815022304))
    y
    x)
   (+
    x
    (fma
     0.0692910599291889
     y
     (-
      (/
       (fma
        (/
         (-
          (* 0.279195317918525 y)
          (fma
           -6.012459259764103
           (* y -0.07512208616047561)
           (* 0.23214887407009785 y)))
         z)
        -1.0
        (* y -0.07512208616047561))
       z))))))

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))) <= 5e+307) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((6.012459259764103 + z), z, 3.350343815022304)), y, x);
	} else {
		tmp = x + fma(0.0692910599291889, y, -(fma((((0.279195317918525 * y) - fma(-6.012459259764103, (y * -0.07512208616047561), (0.23214887407009785 * y))) / z), -1.0, (y * -0.07512208616047561)) / z));
	}
	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))) <= 5e+307)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), y, x);
	else
		tmp = Float64(x + fma(0.0692910599291889, y, Float64(-Float64(fma(Float64(Float64(Float64(0.279195317918525 * y) - fma(-6.012459259764103, Float64(y * -0.07512208616047561), Float64(0.23214887407009785 * y))) / z), -1.0, Float64(y * -0.07512208616047561)) / z))));
	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], 5e+307], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y + (-N[(N[(N[(N[(N[(0.279195317918525 * y), $MachinePrecision] - N[(-6.012459259764103 * N[(y * -0.07512208616047561), $MachinePrecision] + N[(0.23214887407009785 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -1.0 + N[(y * -0.07512208616047561), $MachinePrecision]), $MachinePrecision] / z), $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)))) < 5e307

   1. Initial program 95.8%

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

   3. Applied rewrites 99.7%

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

   if 5e307 < (+.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 1.0%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{11167812716741}{40000000000000} \cdot y - \left(\frac{-6012459259764103}{1000000000000000} \cdot \left(\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \frac{72546523146905574025723165383}{312500000000000000000000000000} \cdot y\right)}{z} + \frac{-307332350656623}{625000000000000} \cdot y\right) - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{11167812716741}{40000000000000} \cdot y - \left(\frac{-6012459259764103}{1000000000000000} \cdot \left(\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \frac{72546523146905574025723165383}{312500000000000000000000000000} \cdot y\right)}{z} + \frac{-307332350656623}{625000000000000} \cdot y\right) - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, \color{blue}{y}, -1 \cdot \frac{\left(-1 \cdot \frac{\frac{11167812716741}{40000000000000} \cdot y - \left(\frac{-6012459259764103}{1000000000000000} \cdot \left(\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \frac{72546523146905574025723165383}{312500000000000000000000000000} \cdot y\right)}{z} + \frac{-307332350656623}{625000000000000} \cdot y\right) - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{11167812716741}{40000000000000} \cdot y - \left(\frac{-6012459259764103}{1000000000000000} \cdot \left(\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \frac{72546523146905574025723165383}{312500000000000000000000000000} \cdot y\right)}{z} + \frac{-307332350656623}{625000000000000} \cdot y\right) - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, -\frac{\left(-1 \cdot \frac{\frac{11167812716741}{40000000000000} \cdot y - \left(\frac{-6012459259764103}{1000000000000000} \cdot \left(\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \frac{72546523146905574025723165383}{312500000000000000000000000000} \cdot y\right)}{z} + \frac{-307332350656623}{625000000000000} \cdot y\right) - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, -\frac{\left(-1 \cdot \frac{\frac{11167812716741}{40000000000000} \cdot y - \left(\frac{-6012459259764103}{1000000000000000} \cdot \left(\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \frac{72546523146905574025723165383}{312500000000000000000000000000} \cdot y\right)}{z} + \frac{-307332350656623}{625000000000000} \cdot y\right) - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 99.5%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, -\frac{\mathsf{fma}\left(\frac{0.279195317918525 \cdot y - \mathsf{fma}\left(-6.012459259764103, y \cdot -0.07512208616047561, 0.23214887407009785 \cdot y\right)}{z}, -1, y \cdot -0.07512208616047561\right)}{z}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{+78}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;t\_0 \leq 3 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+285}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (fma 0.0692910599291889 y x)
     (if (<= t_0 -4e+78)
       (* 0.08333333333333323 y)
       (if (<= t_0 3e+86)
         (fma 0.0692910599291889 y x)
         (if (<= t_0 1e+285)
           (* 0.08333333333333323 y)
           (fma 0.0692910599291889 y x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (t_0 <= -4e+78) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 3e+86) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (t_0 <= 1e+285) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(0.0692910599291889, y, x);
	elseif (t_0 <= -4e+78)
		tmp = Float64(0.08333333333333323 * y);
	elseif (t_0 <= 3e+86)
		tmp = fma(0.0692910599291889, y, x);
	elseif (t_0 <= 1e+285)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[t$95$0, -4e+78], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[t$95$0, 3e+86], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 1e+285], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0 or -4.00000000000000003e78 < (/.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))) < 2.99999999999999977e86 or 9.9999999999999998e284 < (/.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 60.6%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 89.1

         \[\leadsto \mathsf{fma}\left(0.0692910599291889, \color{blue}{y}, x\right) \]

   4. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -inf.0 < (/.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))) < -4.00000000000000003e78 or 2.99999999999999977e86 < (/.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))) < 9.9999999999999998e284

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{279195317918525}{3350343815022304} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 89.0

         \[\leadsto \mathsf{fma}\left(0.08333333333333323, \color{blue}{y}, x\right) \]

   4. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 69.0

         \[\leadsto 0.08333333333333323 \cdot y \]

   7. Applied rewrites 69.0%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      1e+285)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (+ 6.012459259764103 z) z 3.350343815022304))
    y
    x)
   (+ (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) x)))

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))) <= 1e+285) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((6.012459259764103 + z), z, 3.350343815022304)), y, x);
	} else {
		tmp = (((0.07512208616047561 / z) + 0.0692910599291889) * y) + x;
	}
	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))) <= 1e+285)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), y, x);
	else
		tmp = Float64(Float64(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889) * y) + x);
	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], 1e+285], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] + x), $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)))) < 9.9999999999999998e284

   1. Initial program 95.7%

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

   3. Applied rewrites 99.7%

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

   if 9.9999999999999998e284 < (+.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 8.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 95.9

         \[\leadsto x + \mathsf{fma}\left(0.0692910599291889, y, \frac{y \cdot 0.07512208616047561}{z}\right) \]

   4. Applied rewrites 95.9%

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

   6. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y + x\\ \mathbf{if}\;z \leq -6.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.00033:\\ \;\;\;\;x + \mathsf{fma}\left(y \cdot -0.00277777777751721, z, 0.08333333333333323 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) x)))
   (if (<= z -6.6)
     t_0
     (if (<= z 0.00033)
       (+ x (fma (* y -0.00277777777751721) z (* 0.08333333333333323 y)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) + x;
	double tmp;
	if (z <= -6.6) {
		tmp = t_0;
	} else if (z <= 0.00033) {
		tmp = x + fma((y * -0.00277777777751721), z, (0.08333333333333323 * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889) * y) + x)
	tmp = 0.0
	if (z <= -6.6)
		tmp = t_0;
	elseif (z <= 0.00033)
		tmp = Float64(x + fma(Float64(y * -0.00277777777751721), z, Float64(0.08333333333333323 * y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -6.6], t$95$0, If[LessEqual[z, 0.00033], N[(x + N[(N[(y * -0.00277777777751721), $MachinePrecision] * z + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5999999999999996 or 3.3e-4 < z

   1. Initial program 38.4%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 98.8

         \[\leadsto x + \mathsf{fma}\left(0.0692910599291889, y, \frac{y \cdot 0.07512208616047561}{z}\right) \]

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.8%

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

   if -6.5999999999999996 < z < 3.3e-4

   1. Initial program 99.7%

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

   2. 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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto x + \mathsf{fma}\left(y \cdot \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      7. lower-*.f64 99.6

         \[\leadsto x + \mathsf{fma}\left(y \cdot -0.00277777777751721, z, 0.08333333333333323 \cdot y\right) \]

   4. Applied rewrites 99.6%

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

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 0.00033:\\ \;\;\;\;x + \mathsf{fma}\left(y \cdot -0.00277777777751721, z, 0.08333333333333323 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.6)
   (fma 0.0692910599291889 y x)
   (if (<= z 0.00033)
     (+ x (fma (* y -0.00277777777751721) z (* 0.08333333333333323 y)))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.6) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 0.00033) {
		tmp = x + fma((y * -0.00277777777751721), z, (0.08333333333333323 * y));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.6)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 0.00033)
		tmp = Float64(x + fma(Float64(y * -0.00277777777751721), z, Float64(0.08333333333333323 * y)));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.6], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 0.00033], N[(x + N[(N[(y * -0.00277777777751721), $MachinePrecision] * z + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5999999999999996 or 3.3e-4 < z

   1. Initial program 38.4%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 98.3

         \[\leadsto \mathsf{fma}\left(0.0692910599291889, \color{blue}{y}, x\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -6.5999999999999996 < z < 3.3e-4

   1. Initial program 99.7%

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

   2. 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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto x + \mathsf{fma}\left(y \cdot \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      7. lower-*.f64 99.6

         \[\leadsto x + \mathsf{fma}\left(y \cdot -0.00277777777751721, z, 0.08333333333333323 \cdot y\right) \]

   4. Applied rewrites 99.6%

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

Alternative 6: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 0.00033:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.6)
   (fma 0.0692910599291889 y x)
   (if (<= z 0.00033)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.6) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 0.00033) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.6)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 0.00033)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.6], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 0.00033], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5999999999999996 or 3.3e-4 < z

   1. Initial program 38.4%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 98.3

         \[\leadsto \mathsf{fma}\left(0.0692910599291889, \color{blue}{y}, x\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -6.5999999999999996 < z < 3.3e-4

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{279195317918525}{3350343815022304} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(0.08333333333333323, \color{blue}{y}, x\right) \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

      2. flip-+ N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

      3. unpow2 N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

      4. metadata-eval N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

   7. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot \left(y \cdot z\right) + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-182}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e-96)
   x
   (if (<= x 2.5e-182)
     (* 0.08333333333333323 y)
     (if (<= x 1.8e-21) (* 0.0692910599291889 y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-96) {
		tmp = x;
	} else if (x <= 2.5e-182) {
		tmp = 0.08333333333333323 * y;
	} else if (x <= 1.8e-21) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = x;
	}
	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 (x <= (-9.5d-96)) then
        tmp = x
    else if (x <= 2.5d-182) then
        tmp = 0.08333333333333323d0 * y
    else if (x <= 1.8d-21) then
        tmp = 0.0692910599291889d0 * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-96) {
		tmp = x;
	} else if (x <= 2.5e-182) {
		tmp = 0.08333333333333323 * y;
	} else if (x <= 1.8e-21) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e-96:
		tmp = x
	elif x <= 2.5e-182:
		tmp = 0.08333333333333323 * y
	elif x <= 1.8e-21:
		tmp = 0.0692910599291889 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e-96)
		tmp = x;
	elseif (x <= 2.5e-182)
		tmp = Float64(0.08333333333333323 * y);
	elseif (x <= 1.8e-21)
		tmp = Float64(0.0692910599291889 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e-96)
		tmp = x;
	elseif (x <= 2.5e-182)
		tmp = 0.08333333333333323 * y;
	elseif (x <= 1.8e-21)
		tmp = 0.0692910599291889 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e-96], x, If[LessEqual[x, 2.5e-182], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[x, 1.8e-21], N[(0.0692910599291889 * y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999993e-96 or 1.79999999999999995e-21 < x

   1. Initial program 68.8%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 69.4%

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

   if -9.4999999999999993e-96 < x < 2.50000000000000012e-182

   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

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

      1. +-commutative N/A

         \[\leadsto \frac{279195317918525}{3350343815022304} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 65.5

         \[\leadsto \mathsf{fma}\left(0.08333333333333323, \color{blue}{y}, x\right) \]

   4. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 50.2

         \[\leadsto 0.08333333333333323 \cdot y \]

   7. Applied rewrites 50.2%

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

   if 2.50000000000000012e-182 < x < 1.79999999999999995e-21

   1. Initial program 67.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{279195317918525}{3350343815022304} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 71.9

         \[\leadsto \mathsf{fma}\left(0.08333333333333323, \color{blue}{y}, x\right) \]

   4. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

      2. flip-+ N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

      3. unpow2 N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

      4. metadata-eval N/A

         \[\leadsto x + \frac{692910599291889}{10000000000000000} \cdot y \]

   7. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 40.5

         \[\leadsto 0.0692910599291889 \cdot y \]

   10. Applied rewrites 40.5%

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

Alternative 8: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 0.00033:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.6)
   (fma 0.0692910599291889 y x)
   (if (<= z 0.00033)
     (fma 0.08333333333333323 y x)
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.6) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 0.00033) {
		tmp = fma(0.08333333333333323, y, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.6)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 0.00033)
		tmp = fma(0.08333333333333323, y, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.6], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 0.00033], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5999999999999996 or 3.3e-4 < z

   1. Initial program 38.4%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 98.3

         \[\leadsto \mathsf{fma}\left(0.0692910599291889, \color{blue}{y}, x\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -6.5999999999999996 < z < 3.3e-4

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{279195317918525}{3350343815022304} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(0.08333333333333323, \color{blue}{y}, x\right) \]

   4. Applied rewrites 99.2%

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

Alternative 9: 60.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-58}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e-96) x (if (<= x 8e-58) (* 0.08333333333333323 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-96) {
		tmp = x;
	} else if (x <= 8e-58) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = x;
	}
	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 (x <= (-9.5d-96)) then
        tmp = x
    else if (x <= 8d-58) then
        tmp = 0.08333333333333323d0 * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-96) {
		tmp = x;
	} else if (x <= 8e-58) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e-96:
		tmp = x
	elif x <= 8e-58:
		tmp = 0.08333333333333323 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e-96)
		tmp = x;
	elseif (x <= 8e-58)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e-96)
		tmp = x;
	elseif (x <= 8e-58)
		tmp = 0.08333333333333323 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e-96], x, If[LessEqual[x, 8e-58], N[(0.08333333333333323 * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999993e-96 or 8.0000000000000002e-58 < x

   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 x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 68.1%

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

   if -9.4999999999999993e-96 < x < 8.0000000000000002e-58

   1. Initial program 68.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{279195317918525}{3350343815022304} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 67.0

         \[\leadsto \mathsf{fma}\left(0.08333333333333323, \color{blue}{y}, x\right) \]

   4. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 48.0

         \[\leadsto 0.08333333333333323 \cdot y \]

   7. Applied rewrites 48.0%

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

Alternative 10: 50.8% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 68.6%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 50.8%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025095 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -324806146098267/40000000) (- (* (+ (/ 7512208616047561/100000000000000000 z) 692910599291889/10000000000000000) y) (- (/ (* 323697630959937/800000000000000 y) (* z z)) x)) (if (< z 657611897278737700000) (+ x (* (* y (+ (* (+ (* z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000)) (/ 1 (+ (* (+ z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)))) (- (* (+ (/ 7512208616047561/100000000000000000 z) 692910599291889/10000000000000000) y) (- (/ (* 323697630959937/800000000000000 y) (* z z)) x)))))

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