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

Percentage Accurate: 68.4% → 99.7%

Time: 10.2s

Alternatives: 15

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ t_1 := y \cdot 0.0692910599291889 - x\\ \mathbf{if}\;z \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 900000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot y}{t\_0}, \mathsf{fma}\left(y, \frac{0.279195317918525}{t\_0}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_1}, \left(-x\right) \cdot \frac{x}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ 6.012459259764103 z) z 3.350343815022304))
        (t_1 (- (* y 0.0692910599291889) x)))
   (if (<= z -1e+20)
     (fma 0.0692910599291889 y x)
     (if (<= z 900000000000.0)
       (fma
        z
        (/ (* (fma 0.0692910599291889 z 0.4917317610505968) y) t_0)
        (fma y (/ 0.279195317918525 t_0) x))
       (fma (* 0.004801250986110448 y) (/ y t_1) (* (- x) (/ x t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double t_1 = (y * 0.0692910599291889) - x;
	double tmp;
	if (z <= -1e+20) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 900000000000.0) {
		tmp = fma(z, ((fma(0.0692910599291889, z, 0.4917317610505968) * y) / t_0), fma(y, (0.279195317918525 / t_0), x));
	} else {
		tmp = fma((0.004801250986110448 * y), (y / t_1), (-x * (x / t_1)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	t_1 = Float64(Float64(y * 0.0692910599291889) - x)
	tmp = 0.0
	if (z <= -1e+20)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 900000000000.0)
		tmp = fma(z, Float64(Float64(fma(0.0692910599291889, z, 0.4917317610505968) * y) / t_0), fma(y, Float64(0.279195317918525 / t_0), x));
	else
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_1), Float64(Float64(-x) * Float64(x / t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * 0.0692910599291889), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -1e+20], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 900000000000.0], N[(z * N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * y), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(y * N[(0.279195317918525 / t$95$0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision] + N[((-x) * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e20

   1. Initial program 33.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1e20 < z < 9e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   if 9e11 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 51.6%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

Alternative 2: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+78} \lor \neg \left(t\_0 \leq 10^{+184} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+294}\right)\right):\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* 0.0692910599291889 y)
     (if (or (<= t_0 -2e+78) (not (or (<= t_0 1e+184) (not (<= t_0 5e+294)))))
       (* 0.08333333333333323 y)
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0692910599291889 * y;
	} else if ((t_0 <= -2e+78) || !((t_0 <= 1e+184) || !(t_0 <= 5e+294))) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0692910599291889 * y);
	elseif ((t_0 <= -2e+78) || !((t_0 <= 1e+184) || !(t_0 <= 5e+294)))
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y), $MachinePrecision], If[Or[LessEqual[t$95$0, -2e+78], N[Not[Or[LessEqual[t$95$0, 1e+184], N[Not[LessEqual[t$95$0, 5e+294]], $MachinePrecision]]], $MachinePrecision]], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0

   1. Initial program 6.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.5%

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

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -2.00000000000000002e78 or 1.00000000000000002e184 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 4.9999999999999999e294

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.7%

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

   if -2.00000000000000002e78 < (/.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))) < 1.00000000000000002e184 or 4.9999999999999999e294 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 66.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -2 \cdot 10^{+78} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+184} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+294}\right)\right):\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-91}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{+55}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* 0.0692910599291889 y)
     (if (<= t_0 -2e-91)
       (* 0.08333333333333323 y)
       (if (<= t_0 1e+55)
         (* 1.0 x)
         (if (<= t_0 5e+294)
           (* 0.08333333333333323 y)
           (* 0.0692910599291889 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0692910599291889 * y;
	} else if (t_0 <= -2e-91) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 1e+55) {
		tmp = 1.0 * x;
	} else if (t_0 <= 5e+294) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0692910599291889 * y;
	} else if (t_0 <= -2e-91) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 1e+55) {
		tmp = 1.0 * x;
	} else if (t_0 <= 5e+294) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.0692910599291889 * y
	elif t_0 <= -2e-91:
		tmp = 0.08333333333333323 * y
	elif t_0 <= 1e+55:
		tmp = 1.0 * x
	elif t_0 <= 5e+294:
		tmp = 0.08333333333333323 * y
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0692910599291889 * y);
	elseif (t_0 <= -2e-91)
		tmp = Float64(0.08333333333333323 * y);
	elseif (t_0 <= 1e+55)
		tmp = Float64(1.0 * x);
	elseif (t_0 <= 5e+294)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.0692910599291889 * y;
	elseif (t_0 <= -2e-91)
		tmp = 0.08333333333333323 * y;
	elseif (t_0 <= 1e+55)
		tmp = 1.0 * x;
	elseif (t_0 <= 5e+294)
		tmp = 0.08333333333333323 * y;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y), $MachinePrecision], If[LessEqual[t$95$0, -2e-91], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[t$95$0, 1e+55], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+294], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0 or 4.9999999999999999e294 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 1.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.6%

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

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -2.00000000000000004e-91 or 1.00000000000000001e55 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 4.9999999999999999e294

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.4%

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

   if -2.00000000000000004e-91 < (/.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))) < 1.00000000000000001e55

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 93.6

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{279195317918525}{3350343815022304} \cdot \frac{y}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x}, 0.08333333333333323, 1\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 85.4%

       \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ t_1 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{z} \cdot 0.0692910599291889, \mathsf{fma}\left(y, \frac{0.279195317918525}{t\_1}, x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          x
          (/
           (*
            y
            (+
             (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
             0.279195317918525))
           (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
        (t_1 (fma (+ 6.012459259764103 z) z 3.350343815022304)))
   (if (<= t_0 (- INFINITY))
     (fma z (* (/ y z) 0.0692910599291889) (fma y (/ 0.279195317918525 t_1) x))
     (if (<= t_0 5e+294)
       (+
        x
        (/
         (*
          (fma
           (fma 0.0692910599291889 z 0.4917317610505968)
           z
           0.279195317918525)
          y)
         t_1))
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
	double t_1 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(z, ((y / z) * 0.0692910599291889), fma(y, (0.279195317918525 / t_1), x));
	} else if (t_0 <= 5e+294) {
		tmp = x + ((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / t_1);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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)))
	t_1 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(z, Float64(Float64(y / z) * 0.0692910599291889), fma(y, Float64(0.279195317918525 / t_1), x));
	elseif (t_0 <= 5e+294)
		tmp = Float64(x + Float64(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / t_1));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(z * N[(N[(y / z), $MachinePrecision] * 0.0692910599291889), $MachinePrecision] + N[(y * N[(0.279195317918525 / t$95$1), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+294], N[(x + N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < -inf.0

   1. Initial program 6.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 6.4

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

   4. Applied rewrites 6.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 38.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 4.9999999999999999e294

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 4.9999999999999999e294 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 1.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 5: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ t_1 := y \cdot 0.0692910599291889 - x\\ \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot z, 0.004801250986110448, -0.24180012482592123\right) \cdot \frac{y}{t\_0}}{0.0692910599291889 \cdot z - 0.4917317610505968}, \mathsf{fma}\left(y, \frac{0.279195317918525}{t\_0}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_1}, \left(-x\right) \cdot \frac{x}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ 6.012459259764103 z) z 3.350343815022304))
        (t_1 (- (* y 0.0692910599291889) x)))
   (if (<=
        (+
         x
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
        5e+294)
     (fma
      z
      (/
       (* (fma (* z z) 0.004801250986110448 -0.24180012482592123) (/ y t_0))
       (- (* 0.0692910599291889 z) 0.4917317610505968))
      (fma y (/ 0.279195317918525 t_0) x))
     (fma (* 0.004801250986110448 y) (/ y t_1) (* (- x) (/ x t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double t_1 = (y * 0.0692910599291889) - x;
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 5e+294) {
		tmp = fma(z, ((fma((z * z), 0.004801250986110448, -0.24180012482592123) * (y / t_0)) / ((0.0692910599291889 * z) - 0.4917317610505968)), fma(y, (0.279195317918525 / t_0), x));
	} else {
		tmp = fma((0.004801250986110448 * y), (y / t_1), (-x * (x / t_1)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	t_1 = Float64(Float64(y * 0.0692910599291889) - x)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 5e+294)
		tmp = fma(z, Float64(Float64(fma(Float64(z * z), 0.004801250986110448, -0.24180012482592123) * Float64(y / t_0)) / Float64(Float64(0.0692910599291889 * z) - 0.4917317610505968)), fma(y, Float64(0.279195317918525 / t_0), x));
	else
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_1), Float64(Float64(-x) * Float64(x / t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * 0.0692910599291889), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+294], N[(z * N[(N[(N[(N[(z * z), $MachinePrecision] * 0.004801250986110448 + -0.24180012482592123), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0692910599291889 * z), $MachinePrecision] - 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.279195317918525 / t$95$0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision] + N[((-x) * N[(x / t$95$1), $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)))) < 4.9999999999999999e294

   1. Initial program 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. swap-sqr N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\left(\left(z \cdot z\right) \cdot \color{blue}{\frac{480125098611044764748221188321}{100000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{307332350656623}{625000000000000}\right) \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{\mathsf{fma}\left(z \cdot z, \frac{480125098611044764748221188321}{100000000000000000000000000000000}, \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{307332350656623}{625000000000000}\right)} \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{480125098611044764748221188321}{100000000000000000000000000000000}, \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{307332350656623}{625000000000000}\right) \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot z, \frac{480125098611044764748221188321}{100000000000000000000000000000000}, \color{blue}{\frac{-307332350656623}{625000000000000}} \cdot \frac{307332350656623}{625000000000000}\right) \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot z, \frac{480125098611044764748221188321}{100000000000000000000000000000000}, \color{blue}{\frac{-94453173760125479739253764129}{390625000000000000000000000000}}\right) \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot z, \frac{480125098611044764748221188321}{100000000000000000000000000000000}, \frac{-94453173760125479739253764129}{390625000000000000000000000000}\right) \cdot \color{blue}{\frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      18. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot z, \frac{480125098611044764748221188321}{100000000000000000000000000000000}, \frac{-94453173760125479739253764129}{390625000000000000000000000000}\right) \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}}{\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} - \frac{307332350656623}{625000000000000}}}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

   8. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, 0.004801250986110448, -0.24180012482592123\right) \cdot \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}}{0.0692910599291889 \cdot z - 0.4917317610505968}}, \mathsf{fma}\left(y, \frac{0.279195317918525}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\right) \]

   if 4.9999999999999999e294 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 1.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 44.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 44.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

       \[\leadsto \mathsf{fma}\left(0.004801250986110448 \cdot y, \color{blue}{\frac{y}{y \cdot 0.0692910599291889 - x}}, \left(-x\right) \cdot \frac{x}{y \cdot 0.0692910599291889 - x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot 0.0692910599291889 - x\\ t_1 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), \frac{y}{t\_1} \cdot z, \mathsf{fma}\left(\frac{0.279195317918525}{t\_1}, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_0}, \left(-x\right) \cdot \frac{x}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y 0.0692910599291889) x))
        (t_1 (fma (+ 6.012459259764103 z) z 3.350343815022304)))
   (if (<=
        (+
         x
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
        5e+294)
     (fma
      (fma 0.0692910599291889 z 0.4917317610505968)
      (* (/ y t_1) z)
      (fma (/ 0.279195317918525 t_1) y x))
     (fma (* 0.004801250986110448 y) (/ y t_0) (* (- x) (/ x t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * 0.0692910599291889) - x;
	double t_1 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 5e+294) {
		tmp = fma(fma(0.0692910599291889, z, 0.4917317610505968), ((y / t_1) * z), fma((0.279195317918525 / t_1), y, x));
	} else {
		tmp = fma((0.004801250986110448 * y), (y / t_0), (-x * (x / t_0)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * 0.0692910599291889) - x)
	t_1 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 5e+294)
		tmp = fma(fma(0.0692910599291889, z, 0.4917317610505968), Float64(Float64(y / t_1) * z), fma(Float64(0.279195317918525 / t_1), y, x));
	else
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_0), Float64(Float64(-x) * Float64(x / t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * 0.0692910599291889), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+294], N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * N[(N[(y / t$95$1), $MachinePrecision] * z), $MachinePrecision] + N[(N[(0.279195317918525 / t$95$1), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision] + N[((-x) * N[(x / t$95$0), $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)))) < 4.9999999999999999e294

   1. Initial program 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-add N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 97.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 4.9999999999999999e294 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 1.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 44.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 44.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

       \[\leadsto \mathsf{fma}\left(0.004801250986110448 \cdot y, \color{blue}{\frac{y}{y \cdot 0.0692910599291889 - x}}, \left(-x\right) \cdot \frac{x}{y \cdot 0.0692910599291889 - x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot 0.0692910599291889 - x\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_0}, \left(-x\right) \cdot \frac{x}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y 0.0692910599291889) x)))
   (if (<= z -1.35e+14)
     (fma 0.0692910599291889 y x)
     (if (<= z 4.6e+24)
       (+
        x
        (/
         (*
          (fma
           (fma 0.0692910599291889 z 0.4917317610505968)
           z
           0.279195317918525)
          y)
         (fma (+ 6.012459259764103 z) z 3.350343815022304)))
       (fma (* 0.004801250986110448 y) (/ y t_0) (* (- x) (/ x t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * 0.0692910599291889) - x;
	double tmp;
	if (z <= -1.35e+14) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 4.6e+24) {
		tmp = x + ((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / fma((6.012459259764103 + z), z, 3.350343815022304));
	} else {
		tmp = fma((0.004801250986110448 * y), (y / t_0), (-x * (x / t_0)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * 0.0692910599291889) - x)
	tmp = 0.0
	if (z <= -1.35e+14)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 4.6e+24)
		tmp = Float64(x + Float64(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)));
	else
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_0), Float64(Float64(-x) * Float64(x / t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * 0.0692910599291889), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -1.35e+14], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 4.6e+24], N[(x + N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * y), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision] + N[((-x) * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e14

   1. Initial program 33.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.35e14 < z < 4.5999999999999998e24

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 4.5999999999999998e24 < z

   1. Initial program 35.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.6%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

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

Alternative 8: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot 0.0692910599291889 - x\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.004801250986110448 \cdot y}{t\_0}, \left(-x\right) \cdot \frac{x}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y 0.0692910599291889) x)))
   (if (<= z -1.35e+14)
     (fma 0.0692910599291889 y x)
     (if (<= z 5e+24)
       (+
        x
        (/
         (*
          (fma
           (fma 0.0692910599291889 z 0.4917317610505968)
           z
           0.279195317918525)
          y)
         (fma (+ 6.012459259764103 z) z 3.350343815022304)))
       (fma y (/ (* 0.004801250986110448 y) t_0) (* (- x) (/ x t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * 0.0692910599291889) - x;
	double tmp;
	if (z <= -1.35e+14) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 5e+24) {
		tmp = x + ((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / fma((6.012459259764103 + z), z, 3.350343815022304));
	} else {
		tmp = fma(y, ((0.004801250986110448 * y) / t_0), (-x * (x / t_0)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * 0.0692910599291889) - x)
	tmp = 0.0
	if (z <= -1.35e+14)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 5e+24)
		tmp = Float64(x + Float64(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)));
	else
		tmp = fma(y, Float64(Float64(0.004801250986110448 * y) / t_0), Float64(Float64(-x) * Float64(x / t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * 0.0692910599291889), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -1.35e+14], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 5e+24], N[(x + N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * y), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(0.004801250986110448 * y), $MachinePrecision] / t$95$0), $MachinePrecision] + N[((-x) * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e14

   1. Initial program 33.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.35e14 < z < 5.00000000000000045e24

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 5.00000000000000045e24 < z

   1. Initial program 35.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

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

Alternative 9: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+14} \lor \neg \left(z \leq 4.6 \cdot 10^{+24}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.35e+14) (not (<= z 4.6e+24)))
   (fma 0.0692910599291889 y x)
   (+
    x
    (/
     (*
      (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
      y)
     (fma (+ 6.012459259764103 z) z 3.350343815022304)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.35e+14) || !(z <= 4.6e+24)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = x + ((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / fma((6.012459259764103 + z), z, 3.350343815022304));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.35e+14) || !(z <= 4.6e+24))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = Float64(x + Float64(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * y) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.35e+14], N[Not[LessEqual[z, 4.6e+24]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(x + N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * y), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e14 or 4.5999999999999998e24 < z

   1. Initial program 34.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1.35e14 < z < 4.5999999999999998e24

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+14} \lor \neg \left(z \leq 4.6 \cdot 10^{+24}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, -0.004191293246138338, y \cdot 0.004984943827291682\right), z, -0.00277777777751721 \cdot y\right), z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -53000.0) (not (<= z 8e-6)))
   (fma 0.07512208616047561 (/ y z) (fma 0.0692910599291889 y x))
   (fma
    (fma
     (fma y -0.004191293246138338 (* y 0.004984943827291682))
     z
     (* -0.00277777777751721 y))
    z
    (fma 0.08333333333333323 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -53000.0) || !(z <= 8e-6)) {
		tmp = fma(0.07512208616047561, (y / z), fma(0.0692910599291889, y, x));
	} else {
		tmp = fma(fma(fma(y, -0.004191293246138338, (y * 0.004984943827291682)), z, (-0.00277777777751721 * y)), z, fma(0.08333333333333323, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -53000.0) || !(z <= 8e-6))
		tmp = fma(0.07512208616047561, Float64(y / z), fma(0.0692910599291889, y, x));
	else
		tmp = fma(fma(fma(y, -0.004191293246138338, Float64(y * 0.004984943827291682)), z, Float64(-0.00277777777751721 * y)), z, fma(0.08333333333333323, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -53000.0], N[Not[LessEqual[z, 8e-6]], $MachinePrecision]], N[(0.07512208616047561 * N[(y / z), $MachinePrecision] + N[(0.0692910599291889 * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -0.004191293246138338 + N[(y * 0.004984943827291682), $MachinePrecision]), $MachinePrecision] * z + N[(-0.00277777777751721 * y), $MachinePrecision]), $MachinePrecision] * z + N[(0.08333333333333323 * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -53000 or 7.99999999999999964e-6 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

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

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sub-sign N/A

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

   5. Applied rewrites 99.4%

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

   if -53000 < z < 7.99999999999999964e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, -0.004191293246138338, y \cdot 0.004984943827291682\right), z, -0.00277777777751721 \cdot y\right), z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, -0.004191293246138338, y \cdot 0.004984943827291682\right), z, -0.00277777777751721 \cdot y\right), z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.00277777777751721 \cdot y, z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -53000.0) (not (<= z 8e-6)))
   (fma 0.07512208616047561 (/ y z) (fma 0.0692910599291889 y x))
   (fma (* -0.00277777777751721 y) z (fma 0.08333333333333323 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -53000.0) || !(z <= 8e-6)) {
		tmp = fma(0.07512208616047561, (y / z), fma(0.0692910599291889, y, x));
	} else {
		tmp = fma((-0.00277777777751721 * y), z, fma(0.08333333333333323, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -53000.0) || !(z <= 8e-6))
		tmp = fma(0.07512208616047561, Float64(y / z), fma(0.0692910599291889, y, x));
	else
		tmp = fma(Float64(-0.00277777777751721 * y), z, fma(0.08333333333333323, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -53000.0], N[Not[LessEqual[z, 8e-6]], $MachinePrecision]], N[(0.07512208616047561 * N[(y / z), $MachinePrecision] + N[(0.0692910599291889 * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.00277777777751721 * y), $MachinePrecision] * z + N[(0.08333333333333323 * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -53000 or 7.99999999999999964e-6 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

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

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sub-sign N/A

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

   5. Applied rewrites 99.4%

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

   if -53000 < z < 7.99999999999999964e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.00277777777751721 \cdot y, z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.00277777777751721 \cdot y, z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -53000.0) (not (<= z 8e-6)))
   (fma 0.0692910599291889 y x)
   (fma (* -0.00277777777751721 y) z (fma 0.08333333333333323 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -53000.0) || !(z <= 8e-6)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma((-0.00277777777751721 * y), z, fma(0.08333333333333323, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -53000.0) || !(z <= 8e-6))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(Float64(-0.00277777777751721 * y), z, fma(0.08333333333333323, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -53000.0], N[Not[LessEqual[z, 8e-6]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(N[(-0.00277777777751721 * y), $MachinePrecision] * z + N[(0.08333333333333323 * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -53000 or 7.99999999999999964e-6 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -53000 < z < 7.99999999999999964e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.00277777777751721 \cdot y, z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -53000.0) (not (<= z 8e-6)))
   (fma 0.0692910599291889 y x)
   (fma 0.08333333333333323 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -53000.0) || !(z <= 8e-6)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(0.08333333333333323, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -53000.0) || !(z <= 8e-6))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(0.08333333333333323, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -53000.0], N[Not[LessEqual[z, 8e-6]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -53000 or 7.99999999999999964e-6 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -53000 < z < 7.99999999999999964e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -53000 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \lor \neg \left(z \leq 1350000000\right):\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.2) (not (<= z 1350000000.0)))
   (* 0.0692910599291889 y)
   (* 0.08333333333333323 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2) || !(z <= 1350000000.0)) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 0.08333333333333323 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.2d0)) .or. (.not. (z <= 1350000000.0d0))) then
        tmp = 0.0692910599291889d0 * y
    else
        tmp = 0.08333333333333323d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2) || !(z <= 1350000000.0)) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 0.08333333333333323 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.2) or not (z <= 1350000000.0):
		tmp = 0.0692910599291889 * y
	else:
		tmp = 0.08333333333333323 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.2) || !(z <= 1350000000.0))
		tmp = Float64(0.0692910599291889 * y);
	else
		tmp = Float64(0.08333333333333323 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.2) || ~((z <= 1350000000.0)))
		tmp = 0.0692910599291889 * y;
	else
		tmp = 0.08333333333333323 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.2], N[Not[LessEqual[z, 1350000000.0]], $MachinePrecision]], N[(0.0692910599291889 * y), $MachinePrecision], N[(0.08333333333333323 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.20000000000000018 or 1.35e9 < z

   1. Initial program 40.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.4%

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

   if -6.20000000000000018 < z < 1.35e9

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \lor \neg \left(z \leq 1350000000\right):\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

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 = 0.0692910599291889d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 80.8

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

5. Applied rewrites 80.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 35.3%

   \[\leadsto 0.0692910599291889 \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025015 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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