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

Percentage Accurate: 68.7% → 99.4%

Time: 4.4s

Alternatives: 12

Speedup: 5.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(6.012459259764103, z, 3.350343815022304\right)\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{14.431876219268936}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      2e+285)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma z z (fma 6.012459259764103 z 3.350343815022304)))
    y
    x)
   (+ x (/ 1.0 (/ 14.431876219268936 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 2e+285) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(z, z, fma(6.012459259764103, z, 3.350343815022304))), y, x);
	} else {
		tmp = x + (1.0 / (14.431876219268936 / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 2e+285)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(z, z, fma(6.012459259764103, z, 3.350343815022304))), y, x);
	else
		tmp = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+285], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(z * z + N[(6.012459259764103 * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 2e285

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-+r+ N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-fma.f64 74.7%

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

   5. Applied rewrites 74.7%

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

   if 2e285 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 68.6%

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.6%

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 68.6%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 68.6%

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 68.6%

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

   3. Applied rewrites 68.6%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 79.7%

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

   6. Applied rewrites 79.7%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{14.431876219268936}{y}}\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      2e+285)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (- z -6.012459259764103) z 3.350343815022304))
    y
    x)
   (+ x (/ 1.0 (/ 14.431876219268936 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 2e+285) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((z - -6.012459259764103), z, 3.350343815022304)), y, x);
	} else {
		tmp = x + (1.0 / (14.431876219268936 / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 2e+285)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(z - -6.012459259764103), z, 3.350343815022304)), y, x);
	else
		tmp = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+285], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(z - -6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 2e285

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   if 2e285 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 68.6%

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.6%

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 68.6%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 68.6%

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 68.6%

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

   3. Applied rewrites 68.6%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 79.7%

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

   6. Applied rewrites 79.7%

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

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.0692910599291889 + -1 \cdot \frac{0.4046220386999212 \cdot \frac{1}{z} - 0.07512208616047561}{z}, y, x\right)\\ \mathbf{if}\;z \leq -6.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (+
           0.0692910599291889
           (*
            -1.0
            (/ (- (* 0.4046220386999212 (/ 1.0 z)) 0.07512208616047561) z)))
          y
          x)))
   (if (<= z -6.2)
     t_0
     (if (<= z 5.0)
       (fma
        (+
         0.08333333333333323
         (* z (- (* 0.0007936505811533442 z) 0.00277777777751721)))
        y
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((0.0692910599291889 + (-1.0 * (((0.4046220386999212 * (1.0 / z)) - 0.07512208616047561) / z))), y, x);
	double tmp;
	if (z <= -6.2) {
		tmp = t_0;
	} else if (z <= 5.0) {
		tmp = fma((0.08333333333333323 + (z * ((0.0007936505811533442 * z) - 0.00277777777751721))), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(0.0692910599291889 + Float64(-1.0 * Float64(Float64(Float64(0.4046220386999212 * Float64(1.0 / z)) - 0.07512208616047561) / z))), y, x)
	tmp = 0.0
	if (z <= -6.2)
		tmp = t_0;
	elseif (z <= 5.0)
		tmp = fma(Float64(0.08333333333333323 + Float64(z * Float64(Float64(0.0007936505811533442 * z) - 0.00277777777751721))), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0692910599291889 + N[(-1.0 * N[(N[(N[(0.4046220386999212 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] - 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -6.2], t$95$0, If[LessEqual[z, 5.0], N[(N[(0.08333333333333323 + N[(z * N[(N[(0.0007936505811533442 * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.20000000000000018 or 5 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{-1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + -1 \cdot \color{blue}{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{\color{blue}{z}}, y, x\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}, y, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}, y, x\right) \]

      6. lower-/.f64 57.9%

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

   6. Applied rewrites 57.9%

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

   if -6.20000000000000018 < z < 5

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + \color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + z \cdot \color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \color{blue}{\frac{155900051080628738716045985239}{56124018394291031809500087342080}}\right), y, x\right) \]

      4. lower-*.f64 58.3%

         \[\leadsto \mathsf{fma}\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right), y, x\right) \]

   6. Applied rewrites 58.3%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-0.4917317610505968 \cdot y - -0.4166096748901212 \cdot y}{z}, 0.0692910599291889 \cdot y\right) + x\\ \mathbf{elif}\;z \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z} - -0.0692910599291889, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5)
   (+
    (fma
     -1.0
     (/ (- (* -0.4917317610505968 y) (* -0.4166096748901212 y)) z)
     (* 0.0692910599291889 y))
    x)
   (if (<= z 4.5)
     (fma
      (+
       0.08333333333333323
       (* z (- (* 0.0007936505811533442 z) 0.00277777777751721)))
      y
      x)
     (fma (- (/ 0.07512208616047561 z) -0.0692910599291889) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5) {
		tmp = fma(-1.0, (((-0.4917317610505968 * y) - (-0.4166096748901212 * y)) / z), (0.0692910599291889 * y)) + x;
	} else if (z <= 4.5) {
		tmp = fma((0.08333333333333323 + (z * ((0.0007936505811533442 * z) - 0.00277777777751721))), y, x);
	} else {
		tmp = fma(((0.07512208616047561 / z) - -0.0692910599291889), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5)
		tmp = Float64(fma(-1.0, Float64(Float64(Float64(-0.4917317610505968 * y) - Float64(-0.4166096748901212 * y)) / z), Float64(0.0692910599291889 * y)) + x);
	elseif (z <= 4.5)
		tmp = fma(Float64(0.08333333333333323 + Float64(z * Float64(Float64(0.0007936505811533442 * z) - 0.00277777777751721))), y, x);
	else
		tmp = fma(Float64(Float64(0.07512208616047561 / z) - -0.0692910599291889), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.5], N[(N[(-1.0 * N[(N[(N[(-0.4917317610505968 * y), $MachinePrecision] - N[(-0.4166096748901212 * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.5], N[(N[(0.08333333333333323 + N[(z * N[(N[(0.0007936505811533442 * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] - -0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.08333333333333323}, y, x\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.9%

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in z around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 65.5%

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

   11. Applied rewrites 65.5%

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

   if -4.5 < z < 4.5

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + \color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + z \cdot \color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \color{blue}{\frac{155900051080628738716045985239}{56124018394291031809500087342080}}\right), y, x\right) \]

      4. lower-*.f64 58.3%

         \[\leadsto \mathsf{fma}\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right), y, x\right) \]

   6. Applied rewrites 58.3%

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

   if 4.5 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \color{blue}{\frac{1}{z}}, y, x\right) \]

      3. lower-/.f64 65.5%

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

   6. Applied rewrites 65.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} + \color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]

      3. add-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right)}, y, x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      5. lower--.f64 65.5%

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

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      8. mult-flip-rev N/A

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

      9. lower-/.f64 65.5%

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

   8. Applied rewrites 65.5%

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

Alternative 5: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{0.07512208616047561}{z} - -0.0692910599291889, y, x\right)\\ \mathbf{if}\;z \leq -4.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (/ 0.07512208616047561 z) -0.0692910599291889) y x)))
   (if (<= z -4.5)
     t_0
     (if (<= z 4.5)
       (fma
        (+
         0.08333333333333323
         (* z (- (* 0.0007936505811533442 z) 0.00277777777751721)))
        y
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(((0.07512208616047561 / z) - -0.0692910599291889), y, x);
	double tmp;
	if (z <= -4.5) {
		tmp = t_0;
	} else if (z <= 4.5) {
		tmp = fma((0.08333333333333323 + (z * ((0.0007936505811533442 * z) - 0.00277777777751721))), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(0.07512208616047561 / z) - -0.0692910599291889), y, x)
	tmp = 0.0
	if (z <= -4.5)
		tmp = t_0;
	elseif (z <= 4.5)
		tmp = fma(Float64(0.08333333333333323 + Float64(z * Float64(Float64(0.0007936505811533442 * z) - 0.00277777777751721))), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] - -0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -4.5], t$95$0, If[LessEqual[z, 4.5], N[(N[(0.08333333333333323 + N[(z * N[(N[(0.0007936505811533442 * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5 or 4.5 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \color{blue}{\frac{1}{z}}, y, x\right) \]

      3. lower-/.f64 65.5%

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

   6. Applied rewrites 65.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} + \color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]

      3. add-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right)}, y, x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      5. lower--.f64 65.5%

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

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      8. mult-flip-rev N/A

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

      9. lower-/.f64 65.5%

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

   8. Applied rewrites 65.5%

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

   if -4.5 < z < 4.5

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + \color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + z \cdot \color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{279195317918525}{3350343815022304} + z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} \cdot z - \color{blue}{\frac{155900051080628738716045985239}{56124018394291031809500087342080}}\right), y, x\right) \]

      4. lower-*.f64 58.3%

         \[\leadsto \mathsf{fma}\left(0.08333333333333323 + z \cdot \left(0.0007936505811533442 \cdot z - 0.00277777777751721\right), y, x\right) \]

   6. Applied rewrites 58.3%

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

Alternative 6: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{0.07512208616047561}{z} - -0.0692910599291889, y, x\right)\\ \mathbf{if}\;z \leq -6.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323 + -0.00277777777751721 \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (/ 0.07512208616047561 z) -0.0692910599291889) y x)))
   (if (<= z -6.8)
     t_0
     (if (<= z 4.0)
       (fma (+ 0.08333333333333323 (* -0.00277777777751721 z)) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(((0.07512208616047561 / z) - -0.0692910599291889), y, x);
	double tmp;
	if (z <= -6.8) {
		tmp = t_0;
	} else if (z <= 4.0) {
		tmp = fma((0.08333333333333323 + (-0.00277777777751721 * z)), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(0.07512208616047561 / z) - -0.0692910599291889), y, x)
	tmp = 0.0
	if (z <= -6.8)
		tmp = t_0;
	elseif (z <= 4.0)
		tmp = fma(Float64(0.08333333333333323 + Float64(-0.00277777777751721 * z)), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] - -0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -6.8], t$95$0, If[LessEqual[z, 4.0], N[(N[(0.08333333333333323 + N[(-0.00277777777751721 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.79999999999999982 or 4 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \color{blue}{\frac{1}{z}}, y, x\right) \]

      3. lower-/.f64 65.5%

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

   6. Applied rewrites 65.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}}, y, x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} + \color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]

      3. add-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \color{blue}{\left(\mathsf{neg}\left(\frac{692910599291889}{10000000000000000}\right)\right)}, y, x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      5. lower--.f64 65.5%

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

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} - \frac{-692910599291889}{10000000000000000}, y, x\right) \]

      8. mult-flip-rev N/A

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

      9. lower-/.f64 65.5%

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

   8. Applied rewrites 65.5%

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

   if -6.79999999999999982 < z < 4

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 64.7%

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

   6. Applied rewrites 64.7%

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

Alternative 7: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -0.92:\\ \;\;\;\;x + \frac{1}{\frac{14.431876219268936}{y}}\\ \mathbf{elif}\;z \leq 1.55:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323 + -0.00277777777751721 \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.92)
   (+ x (/ 1.0 (/ 14.431876219268936 y)))
   (if (<= z 1.55)
     (fma (+ 0.08333333333333323 (* -0.00277777777751721 z)) y x)
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.92) {
		tmp = x + (1.0 / (14.431876219268936 / y));
	} else if (z <= 1.55) {
		tmp = fma((0.08333333333333323 + (-0.00277777777751721 * z)), y, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.92)
		tmp = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)));
	elseif (z <= 1.55)
		tmp = fma(Float64(0.08333333333333323 + Float64(-0.00277777777751721 * z)), y, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.92], N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55], N[(N[(0.08333333333333323 + N[(-0.00277777777751721 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.92000000000000004

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 68.6%

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.6%

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 68.6%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 68.6%

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 68.6%

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

   3. Applied rewrites 68.6%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 79.7%

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

   6. Applied rewrites 79.7%

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

   if -0.92000000000000004 < z < 1.55000000000000004

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   if 1.55000000000000004 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 79.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.0692910599291889}, y, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -470:\\ \;\;\;\;x + \frac{1}{\frac{14.431876219268936}{y}}\\ \mathbf{elif}\;z \leq 210:\\ \;\;\;\;x + \frac{1}{\frac{12.000000000000014}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -470.0)
   (+ x (/ 1.0 (/ 14.431876219268936 y)))
   (if (<= z 210.0)
     (+ x (/ 1.0 (/ 12.000000000000014 y)))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -470.0) {
		tmp = x + (1.0 / (14.431876219268936 / y));
	} else if (z <= 210.0) {
		tmp = x + (1.0 / (12.000000000000014 / y));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -470.0)
		tmp = Float64(x + Float64(1.0 / Float64(14.431876219268936 / y)));
	elseif (z <= 210.0)
		tmp = Float64(x + Float64(1.0 / Float64(12.000000000000014 / y)));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -470.0], N[(x + N[(1.0 / N[(14.431876219268936 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 210.0], N[(x + N[(1.0 / N[(12.000000000000014 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -470

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 68.6%

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.6%

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 68.6%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 68.6%

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 68.6%

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

   3. Applied rewrites 68.6%

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

   4. Taylor expanded in z around inf

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

      1. lower-/.f64 79.7%

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

   6. Applied rewrites 79.7%

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

   if -470 < z < 210

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 68.6%

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.6%

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 68.6%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 68.6%

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 68.6%

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

   3. Applied rewrites 68.6%

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

   4. Taylor expanded in z around 0

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{3350343815022304}{279195317918525}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 78.9%

         \[\leadsto x + \frac{1}{\frac{12.000000000000014}{\color{blue}{y}}} \]

   6. Applied rewrites 78.9%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{12.000000000000014}{y}}} \]

   if 210 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 79.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.0692910599291889}, y, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -470:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 210:\\ \;\;\;\;x + \frac{1}{\frac{12.000000000000014}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -470.0)
   (fma 0.0692910599291889 y x)
   (if (<= z 210.0)
     (+ x (/ 1.0 (/ 12.000000000000014 y)))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -470.0) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 210.0) {
		tmp = x + (1.0 / (12.000000000000014 / y));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -470.0)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 210.0)
		tmp = Float64(x + Float64(1.0 / Float64(12.000000000000014 / y)));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -470.0], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 210.0], N[(x + N[(1.0 / N[(12.000000000000014 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -470 or 210 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 79.6%

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

   if -470 < z < 210

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 68.6%

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 68.6%

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 68.6%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 68.6%

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 68.6%

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

   3. Applied rewrites 68.6%

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

   4. Taylor expanded in z around 0

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{3350343815022304}{279195317918525}}{y}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 78.9%

         \[\leadsto x + \frac{1}{\frac{12.000000000000014}{\color{blue}{y}}} \]

   6. Applied rewrites 78.9%

      \[\leadsto x + \frac{1}{\color{blue}{\frac{12.000000000000014}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -470:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 210:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -470.0)
   (fma 0.0692910599291889 y x)
   (if (<= z 210.0)
     (fma 0.08333333333333323 y x)
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -470.0) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 210.0) {
		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 <= -470.0)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 210.0)
		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, -470.0], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 210.0], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -470 or 210 < z

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 79.6%

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

   if -470 < z < 210

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.9%

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

Alternative 11: 79.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(0.0692910599291889, y, x)
end

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 74.7%

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

4. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]
5. Step-by-step derivation

6. Applied rewrites 79.6%

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

Alternative 12: 50.3% accurate, 7.5× speedup

Math

\[x \cdot 1 \]

FPCore

(FPCore (x y z) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 1.0;
}

Python

def code(x, y, z):
	return x * 1.0

Julia

function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 61.4%

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in x around inf

   \[\leadsto x \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 50.3%

   \[\leadsto x \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(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))))