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

Percentage Accurate: 69.4% → 99.6%

Time: 4.0s

Alternatives: 10

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 69.4% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448\right) \cdot y}{\frac{0.07512208616047561}{z} - 0.0692910599291889} + x\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 270000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (* (- (/ 0.005643327829101921 (* z z)) 0.004801250986110448) y)
           (- (/ 0.07512208616047561 z) 0.0692910599291889))
          x)))
   (if (<= z -1.55e+63)
     t_0
     (if (<= z 270000000.0)
       (fma
        (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
        (/ y (fma (+ 6.012459259764103 z) z 3.350343815022304))
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((((0.005643327829101921 / (z * z)) - 0.004801250986110448) * y) / ((0.07512208616047561 / z) - 0.0692910599291889)) + x;
	double tmp;
	if (z <= -1.55e+63) {
		tmp = t_0;
	} else if (z <= 270000000.0) {
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), (y / fma((6.012459259764103 + z), z, 3.350343815022304)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(0.005643327829101921 / Float64(z * z)) - 0.004801250986110448) * y) / Float64(Float64(0.07512208616047561 / z) - 0.0692910599291889)) + x)
	tmp = 0.0
	if (z <= -1.55e+63)
		tmp = t_0;
	elseif (z <= 270000000.0)
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), Float64(y / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(0.005643327829101921 / N[(z * z), $MachinePrecision]), $MachinePrecision] - 0.004801250986110448), $MachinePrecision] * y), $MachinePrecision] / N[(N[(0.07512208616047561 / z), $MachinePrecision] - 0.0692910599291889), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.55e+63], t$95$0, If[LessEqual[z, 270000000.0], N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * N[(y / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e63 or 2.7e8 < z

   1. Initial program 32.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} + \frac{692910599291889}{10000000000000000}\right) \cdot y + x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} + \frac{692910599291889}{10000000000000000}\right) \cdot y + x \]

      3. flip-+ N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} \cdot \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} \cdot \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} \cdot \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      6. frac-times N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z \cdot z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{{z}^{2}} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{{z}^{2}} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{{z}^{2}} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{480125098611044764748221188321}{100000000000000000000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{480125098611044764748221188321}{100000000000000000000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      14. lift-/.f64 99.4

          \[\leadsto \frac{\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448}{\frac{0.07512208616047561}{z} - 0.0692910599291889} \cdot y + x \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448}{\frac{0.07512208616047561}{z} - 0.0692910599291889} \cdot y + x \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\left(\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448\right) \cdot y}{\color{blue}{\frac{0.07512208616047561}{z} - 0.0692910599291889}} + x \]

   if -1.55e63 < z < 2.7e8

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 98.8

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.8

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

   3. Applied rewrites 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -50000:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y + x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448\right) \cdot y}{\frac{0.07512208616047561}{z} - 0.0692910599291889} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -50000.0)
   (+ (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) x)
   (if (<= z 7e-20)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (+
      (/
       (* (- (/ 0.005643327829101921 (* z z)) 0.004801250986110448) y)
       (- (/ 0.07512208616047561 z) 0.0692910599291889))
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -50000.0) {
		tmp = (((0.07512208616047561 / z) + 0.0692910599291889) * y) + x;
	} else if (z <= 7e-20) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = ((((0.005643327829101921 / (z * z)) - 0.004801250986110448) * y) / ((0.07512208616047561 / z) - 0.0692910599291889)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -50000.0)
		tmp = Float64(Float64(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889) * y) + x);
	elseif (z <= 7e-20)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.005643327829101921 / Float64(z * z)) - 0.004801250986110448) * y) / Float64(Float64(0.07512208616047561 / z) - 0.0692910599291889)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -50000.0], N[(N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7e-20], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(N[(N[(0.005643327829101921 / N[(z * z), $MachinePrecision]), $MachinePrecision] - 0.004801250986110448), $MachinePrecision] * y), $MachinePrecision] / N[(N[(0.07512208616047561 / z), $MachinePrecision] - 0.0692910599291889), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5e4

   1. Initial program 38.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   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 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -5e4 < z < 7.00000000000000007e-20

   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 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 30.1

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

   4. Applied rewrites 30.1%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 30.1

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

   6. Applied rewrites 30.1%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 60.3%

      \[\leadsto 0.0692910599291889 \cdot y + x \]

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 99.3

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

   12. Applied rewrites 99.3%

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

   if 7.00000000000000007e-20 < z

   1. Initial program 42.9%

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

   2. Taylor expanded in z around 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 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} + \frac{692910599291889}{10000000000000000}\right) \cdot y + x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} + \frac{692910599291889}{10000000000000000}\right) \cdot y + x \]

      3. flip-+ N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} \cdot \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} \cdot \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} \cdot \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      6. frac-times N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z \cdot z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{{z}^{2}} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{{z}^{2}} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{{z}^{2}} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{692910599291889}{10000000000000000} \cdot \frac{692910599291889}{10000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{480125098611044764748221188321}{100000000000000000000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\frac{\frac{564332782910192073616717363631302375546337414698500349161489}{100000000000000000000000000000000000000000000000000000000000000}}{z \cdot z} - \frac{480125098611044764748221188321}{100000000000000000000000000000000}}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} - \frac{692910599291889}{10000000000000000}} \cdot y + x \]

      14. lift-/.f64 96.5

          \[\leadsto \frac{\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448}{\frac{0.07512208616047561}{z} - 0.0692910599291889} \cdot y + x \]

   8. Applied rewrites 96.5%

      \[\leadsto \frac{\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448}{\frac{0.07512208616047561}{z} - 0.0692910599291889} \cdot y + x \]

   10. Applied rewrites 96.7%

       \[\leadsto \frac{\left(\frac{0.005643327829101921}{z \cdot z} - 0.004801250986110448\right) \cdot y}{\color{blue}{\frac{0.07512208616047561}{z} - 0.0692910599291889}} + x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y + x\\ \mathbf{if}\;z \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) x)))
   (if (<= z -50000.0)
     t_0
     (if (<= z 7e-20)
       (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
       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 <= -50000.0) {
		tmp = t_0;
	} else if (z <= 7e-20) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} 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 <= -50000.0)
		tmp = t_0;
	elseif (z <= 7e-20)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	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, -50000.0], t$95$0, If[LessEqual[z, 7e-20], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e4 or 7.00000000000000007e-20 < z

   1. Initial program 40.8%

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

   2. Taylor expanded in 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 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 98.1

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

   6. Applied rewrites 98.0%

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

   if -5e4 < z < 7.00000000000000007e-20

   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 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 30.1

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

   4. Applied rewrites 30.1%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 30.1

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

   6. Applied rewrites 30.1%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 60.3%

      \[\leadsto 0.0692910599291889 \cdot y + x \]

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 99.3

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

   12. Applied rewrites 99.3%

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

Alternative 4: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -50000:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -50000.0)
   (fma 0.0692910599291889 y x)
   (if (<= z 7e-20)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (- x (* -0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -50000.0) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 7e-20) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = x - (-0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -50000.0)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 7e-20)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x - Float64(-0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -50000.0], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 7e-20], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(-0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5e4

   1. Initial program 38.5%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if -5e4 < z < 7.00000000000000007e-20

   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 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 30.1

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

   4. Applied rewrites 30.1%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{y \cdot \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}\right) + x} \]

      3. lower-+.f64 30.1

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

   6. Applied rewrites 30.1%

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

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 60.3%

      \[\leadsto 0.0692910599291889 \cdot y + x \]

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. distribute-lft-in N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 99.3

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

   12. Applied rewrites 99.3%

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

   if 7.00000000000000007e-20 < z

   1. Initial program 42.9%

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

   2. Taylor expanded in z around 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 96.6

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

   4. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right) \cdot y} \]

      4. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right) \cdot y} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

Alternative 5: 98.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -50000:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -50000.0)
   (fma 0.0692910599291889 y x)
   (if (<= z 7e-20)
     (fma 0.08333333333333323 y x)
     (- x (* -0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -50000.0) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 7e-20) {
		tmp = fma(0.08333333333333323, y, x);
	} else {
		tmp = x - (-0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -50000.0)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 7e-20)
		tmp = fma(0.08333333333333323, y, x);
	else
		tmp = Float64(x - Float64(-0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -50000.0], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 7e-20], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(x - N[(-0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5e4

   1. Initial program 38.5%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if -5e4 < z < 7.00000000000000007e-20

   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 99.1

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

   4. Applied rewrites 99.1%

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

   if 7.00000000000000007e-20 < z

   1. Initial program 42.9%

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

   2. Taylor expanded in z around 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 96.6

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

   4. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right) \cdot y} \]

      4. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right) \cdot y} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

Alternative 6: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -50000:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -50000.0)
   (fma 0.0692910599291889 y x)
   (if (<= z 7e-20)
     (fma 0.08333333333333323 y x)
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -50000.0) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 7e-20) {
		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 <= -50000.0)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 7e-20)
		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, -50000.0], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 7e-20], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e4 or 7.00000000000000007e-20 < z

   1. Initial program 40.8%

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

   2. Taylor expanded in 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 97.7

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

   4. Applied rewrites 97.7%

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

   if -5e4 < z < 7.00000000000000007e-20

   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 99.1

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

   4. Applied rewrites 99.1%

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

Alternative 7: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+92}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

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 -2e+92)
       (* 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 <= -2e+92) {
		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 <= -2e+92)
		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, -2e+92], 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 -2.0000000000000001e92 < (/.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 66.2%

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

   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 83.6

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

   4. Applied rewrites 83.6%

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

   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))) < -2.0000000000000001e92

   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 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 72.1

         \[\leadsto 0.08333333333333323 \cdot y \]

   7. Applied rewrites 72.1%

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

Alternative 8: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+121}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 10^{+224}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+104)
   (* 0.08333333333333323 y)
   (if (<= y 1.1e+121)
     x
     (if (<= y 1e+224) (* 0.08333333333333323 y) (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+104) {
		tmp = 0.08333333333333323 * y;
	} else if (y <= 1.1e+121) {
		tmp = x;
	} else if (y <= 1e+224) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	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 (y <= (-1.35d+104)) then
        tmp = 0.08333333333333323d0 * y
    else if (y <= 1.1d+121) then
        tmp = x
    else if (y <= 1d+224) then
        tmp = 0.08333333333333323d0 * y
    else
        tmp = 0.0692910599291889d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+104) {
		tmp = 0.08333333333333323 * y;
	} else if (y <= 1.1e+121) {
		tmp = x;
	} else if (y <= 1e+224) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e+104:
		tmp = 0.08333333333333323 * y
	elif y <= 1.1e+121:
		tmp = x
	elif y <= 1e+224:
		tmp = 0.08333333333333323 * y
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+104)
		tmp = Float64(0.08333333333333323 * y);
	elseif (y <= 1.1e+121)
		tmp = x;
	elseif (y <= 1e+224)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e+104)
		tmp = 0.08333333333333323 * y;
	elseif (y <= 1.1e+121)
		tmp = x;
	elseif (y <= 1e+224)
		tmp = 0.08333333333333323 * y;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+104], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[y, 1.1e+121], x, If[LessEqual[y, 1e+224], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.34999999999999992e104 or 1.10000000000000001e121 < y < 9.9999999999999997e223

   1. Initial program 58.8%

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

   2. Taylor expanded in 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 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 48.4

         \[\leadsto 0.08333333333333323 \cdot y \]

   7. Applied rewrites 48.4%

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

   if -1.34999999999999992e104 < y < 1.10000000000000001e121

   1. Initial program 74.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 66.6%

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

   if 9.9999999999999997e223 < y

   1. Initial program 58.8%

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

   2. Taylor expanded in 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 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 48.5

         \[\leadsto 0.0692910599291889 \cdot y \]

   7. Applied rewrites 48.5%

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

Alternative 9: 59.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+224}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+104)
   (* 0.0692910599291889 y)
   (if (<= y 6.6e+224) x (* 0.0692910599291889 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+104) {
		tmp = 0.0692910599291889 * y;
	} else if (y <= 6.6e+224) {
		tmp = x;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	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 (y <= (-1.35d+104)) then
        tmp = 0.0692910599291889d0 * y
    else if (y <= 6.6d+224) then
        tmp = x
    else
        tmp = 0.0692910599291889d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+104) {
		tmp = 0.0692910599291889 * y;
	} else if (y <= 6.6e+224) {
		tmp = x;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e+104:
		tmp = 0.0692910599291889 * y
	elif y <= 6.6e+224:
		tmp = x
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+104)
		tmp = Float64(0.0692910599291889 * y);
	elseif (y <= 6.6e+224)
		tmp = x;
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e+104)
		tmp = 0.0692910599291889 * y;
	elseif (y <= 6.6e+224)
		tmp = x;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+104], N[(0.0692910599291889 * y), $MachinePrecision], If[LessEqual[y, 6.6e+224], x, N[(0.0692910599291889 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.34999999999999992e104 or 6.59999999999999992e224 < y

   1. Initial program 57.8%

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

   2. Taylor expanded in 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 64.4

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

   4. Applied rewrites 64.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 51.4

         \[\leadsto 0.0692910599291889 \cdot y \]

   7. Applied rewrites 51.4%

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

   if -1.34999999999999992e104 < y < 6.59999999999999992e224

   1. Initial program 72.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 61.9%

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

Alternative 10: 51.2% accurate, 29.9× 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 69.4%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025119 
(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))))