Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.1% → 99.8%

Time: 4.5s

Alternatives: 12

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y \cdot \left(z - x\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

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 - x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

Wolfram

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

Initial Program: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y \cdot \left(z - x\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

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 - x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x}{z}, y, y\right)\\ \mathbf{elif}\;y \leq 410:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+22)
   (fma (/ (- x) z) y y)
   (if (<= y 410.0) (/ (fma (- z x) y x) z) (- y (* (/ x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+22) {
		tmp = fma((-x / z), y, y);
	} else if (y <= 410.0) {
		tmp = fma((z - x), y, x) / z;
	} else {
		tmp = y - ((x / z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+22)
		tmp = fma(Float64(Float64(-x) / z), y, y);
	elseif (y <= 410.0)
		tmp = Float64(fma(Float64(z - x), y, x) / z);
	else
		tmp = Float64(y - Float64(Float64(x / z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+22], N[(N[((-x) / z), $MachinePrecision] * y + y), $MachinePrecision], If[LessEqual[y, 410.0], N[(N[(N[(z - x), $MachinePrecision] * y + x), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7e22

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      4. sub-flip N/A

         \[\leadsto \frac{y \cdot \left(z + \left(\mathsf{neg}\left(x\right)\right)\right)}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{z \cdot y + \left(\mathsf{neg}\left(x\right)\right) \cdot y}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(x\right)\right) \cdot y}{z} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(x \cdot y\right)\right)}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(y \cdot x\right)\right)}{z} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(y \cdot x\right)\right)}{z} \]

      10. add-to-fraction-rev N/A

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

      11. distribute-neg-frac N/A

          \[\leadsto y + \left(\mathsf{neg}\left(\frac{y \cdot x}{z}\right)\right) \]

      12. +-commutative N/A

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

      13. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(y \cdot x\right)}{z} + y \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(y \cdot x\right)}{z} + y \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot y\right)}{z} + y \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y}{z} + y \]

      17. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{z} + y \]

      18. mult-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(y \cdot \frac{1}{z}\right) + y \]

      19. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{z} \cdot y\right) + y \]

      20. associate-*r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{z}\right) \cdot y + y \]

      21. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{z}\right)\right) \cdot y + y \]

      22. mult-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{z}\right)\right) \cdot y + y \]

      23. lift-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{z}\right)\right) \cdot y + y \]

      24. lower-fma.f64 N/A

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

   6. Applied rewrites 67.1%

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

   if -7e22 < y < 410

   1. Initial program 88.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 88.1

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

   3. Applied rewrites 88.1%

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

   if 410 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-lft-in N/A

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

      8. add-flip N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lift-/.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y}{z} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right)}{z}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{y \cdot x}{z}\right)\right)\right)\right) \]

      19. distribute-neg-frac2 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot x}{\mathsf{neg}\left(z\right)}\right)\right) \]

      20. distribute-frac-neg N/A

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

      21. frac-2neg N/A

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

      22. lower--.f64 N/A

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

      23. lift-*.f64 N/A

          \[\leadsto y - \frac{y \cdot x}{z} \]

      24. associate-/l* N/A

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

      25. lift-/.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{z}, y, \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 - y}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-269) (fma (/ (- z x) z) y (/ x z)) (fma x (/ (- 1.0 y) z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-269) {
		tmp = fma(((z - x) / z), y, (x / z));
	} else {
		tmp = fma(x, ((1.0 - y) / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e-269)
		tmp = fma(Float64(Float64(z - x) / z), y, Float64(x / z));
	else
		tmp = fma(x, Float64(Float64(1.0 - y) / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e-269], N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-269

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. common-denominator N/A

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

      6. div-add N/A

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

   3. Applied rewrites 93.8%

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

   if 3.9999999999999998e-269 < y

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. associate--l+ N/A

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

      13. add-to-fraction-rev N/A

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

      14. lower-+.f64 N/A

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

      15. sub-div N/A

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

      16. frac-2neg-rev N/A

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

      17. distribute-lft-neg-out N/A

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

      18. frac-2neg-rev N/A

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

      19. remove-double-neg N/A

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

      20. sub-div N/A

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

      21. sub-negate-rev N/A

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

      22. frac-2neg-rev N/A

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

   3. Applied rewrites 95.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + \frac{x - y \cdot x}{z}} \]

      2. lift-/.f64 N/A

         \[\leadsto y + \color{blue}{\frac{x - y \cdot x}{z}} \]

      3. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

         \[\leadsto \color{blue}{\frac{x - y \cdot x}{z} + \frac{y \cdot z}{z}} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} + \frac{y \cdot z}{z} \]

      7. div-sub N/A

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

      8. mult-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

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

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

      21. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+50}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 - y}{z}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+50) (- y (* (/ x z) y)) (fma x (/ (- 1.0 y) z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+50) {
		tmp = y - ((x / z) * y);
	} else {
		tmp = fma(x, ((1.0 - y) / z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e+50)
		tmp = Float64(y - Float64(Float64(x / z) * y));
	else
		tmp = fma(x, Float64(Float64(1.0 - y) / z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e+50], N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0000000000000001e50

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-lft-in N/A

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

      8. add-flip N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lift-/.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y}{z} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right)}{z}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{y \cdot x}{z}\right)\right)\right)\right) \]

      19. distribute-neg-frac2 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot x}{\mathsf{neg}\left(z\right)}\right)\right) \]

      20. distribute-frac-neg N/A

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

      21. frac-2neg N/A

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

      22. lower--.f64 N/A

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

      23. lift-*.f64 N/A

          \[\leadsto y - \frac{y \cdot x}{z} \]

      24. associate-/l* N/A

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

      25. lift-/.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

   if -1.0000000000000001e50 < y

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. associate--l+ N/A

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

      13. add-to-fraction-rev N/A

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

      14. lower-+.f64 N/A

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

      15. sub-div N/A

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

      16. frac-2neg-rev N/A

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

      17. distribute-lft-neg-out N/A

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

      18. frac-2neg-rev N/A

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

      19. remove-double-neg N/A

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

      20. sub-div N/A

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

      21. sub-negate-rev N/A

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

      22. frac-2neg-rev N/A

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

   3. Applied rewrites 95.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + \frac{x - y \cdot x}{z}} \]

      2. lift-/.f64 N/A

         \[\leadsto y + \color{blue}{\frac{x - y \cdot x}{z}} \]

      3. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

         \[\leadsto \color{blue}{\frac{x - y \cdot x}{z} + \frac{y \cdot z}{z}} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} + \frac{y \cdot z}{z} \]

      7. div-sub N/A

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

      8. mult-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

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

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

      21. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.88:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x}{z}, y, y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.88)
   (fma (/ (- x) z) y y)
   (if (<= y 1.0) (/ (fma z y x) z) (- y (* (/ x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.88) {
		tmp = fma((-x / z), y, y);
	} else if (y <= 1.0) {
		tmp = fma(z, y, x) / z;
	} else {
		tmp = y - ((x / z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.88)
		tmp = fma(Float64(Float64(-x) / z), y, y);
	elseif (y <= 1.0)
		tmp = Float64(fma(z, y, x) / z);
	else
		tmp = Float64(y - Float64(Float64(x / z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.88], N[(N[((-x) / z), $MachinePrecision] * y + y), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(z * y + x), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.880000000000000004

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      4. sub-flip N/A

         \[\leadsto \frac{y \cdot \left(z + \left(\mathsf{neg}\left(x\right)\right)\right)}{z} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{z \cdot y + \left(\mathsf{neg}\left(x\right)\right) \cdot y}{z} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(x\right)\right) \cdot y}{z} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(x \cdot y\right)\right)}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(y \cdot x\right)\right)}{z} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot z + \left(\mathsf{neg}\left(y \cdot x\right)\right)}{z} \]

      10. add-to-fraction-rev N/A

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

      11. distribute-neg-frac N/A

          \[\leadsto y + \left(\mathsf{neg}\left(\frac{y \cdot x}{z}\right)\right) \]

      12. +-commutative N/A

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

      13. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(y \cdot x\right)}{z} + y \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(y \cdot x\right)}{z} + y \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot y\right)}{z} + y \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot y}{z} + y \]

      17. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{z} + y \]

      18. mult-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(y \cdot \frac{1}{z}\right) + y \]

      19. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{z} \cdot y\right) + y \]

      20. associate-*r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{z}\right) \cdot y + y \]

      21. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{z}\right)\right) \cdot y + y \]

      22. mult-flip N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{z}\right)\right) \cdot y + y \]

      23. lift-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{z}\right)\right) \cdot y + y \]

      24. lower-fma.f64 N/A

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

   6. Applied rewrites 67.1%

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

   if -0.880000000000000004 < y < 1

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.7%

      \[\leadsto \frac{x + y \cdot \color{blue}{z}}{z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   if 1 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-lft-in N/A

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

      8. add-flip N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lift-/.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y}{z} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right)}{z}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{y \cdot x}{z}\right)\right)\right)\right) \]

      19. distribute-neg-frac2 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot x}{\mathsf{neg}\left(z\right)}\right)\right) \]

      20. distribute-frac-neg N/A

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

      21. frac-2neg N/A

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

      22. lower--.f64 N/A

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

      23. lift-*.f64 N/A

          \[\leadsto y - \frac{y \cdot x}{z} \]

      24. associate-/l* N/A

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

      25. lift-/.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

Alternative 5: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -0.88:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* (/ x z) y))))
   (if (<= y -0.88) t_0 (if (<= y 1.0) (/ (fma z y x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - ((x / z) * y);
	double tmp;
	if (y <= -0.88) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(z, y, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(Float64(x / z) * y))
	tmp = 0.0
	if (y <= -0.88)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(fma(z, y, x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.88], t$95$0, If[LessEqual[y, 1.0], N[(N[(z * y + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.880000000000000004 or 1 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

      7. distribute-lft-in N/A

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

      8. add-flip N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lift-/.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y}{z} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot \left(\mathsf{neg}\left(x\right)\right)}{z}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y \cdot x\right)}{z}\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{y \cdot x}{z}\right)\right)\right)\right) \]

      19. distribute-neg-frac2 N/A

          \[\leadsto y - \left(\mathsf{neg}\left(\frac{y \cdot x}{\mathsf{neg}\left(z\right)}\right)\right) \]

      20. distribute-frac-neg N/A

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

      21. frac-2neg N/A

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

      22. lower--.f64 N/A

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

      23. lift-*.f64 N/A

          \[\leadsto y - \frac{y \cdot x}{z} \]

      24. associate-/l* N/A

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

      25. lift-/.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

   if -0.880000000000000004 < y < 1

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.7%

      \[\leadsto \frac{x + y \cdot \color{blue}{z}}{z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

Alternative 6: 95.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ y z) (- z x))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (/ (fma z y x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y / z) * (z - x);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(z, y, x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 56.1

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

   6. Applied rewrites 56.1%

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

   if -1 < y < 1

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.7%

      \[\leadsto \frac{x + y \cdot \color{blue}{z}}{z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

Alternative 7: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1 - y}{z}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{z}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (- 1.0 y) z))))
   (if (<= x -2.4e+66) t_0 (if (<= x 3e+38) (fma x (/ 1.0 z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((1.0 - y) / z);
	double tmp;
	if (x <= -2.4e+66) {
		tmp = t_0;
	} else if (x <= 3e+38) {
		tmp = fma(x, (1.0 / z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(1.0 - y) / z))
	tmp = 0.0
	if (x <= -2.4e+66)
		tmp = t_0;
	elseif (x <= 3e+38)
		tmp = fma(x, Float64(1.0 / z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e+66], t$95$0, If[LessEqual[x, 3e+38], N[(x * N[(1.0 / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e66 or 3.0000000000000001e38 < x

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. associate--l+ N/A

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

      13. add-to-fraction-rev N/A

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

      14. lower-+.f64 N/A

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

      15. sub-div N/A

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

      16. frac-2neg-rev N/A

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

      17. distribute-lft-neg-out N/A

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

      18. frac-2neg-rev N/A

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

      19. remove-double-neg N/A

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

      20. sub-div N/A

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

      21. sub-negate-rev N/A

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

      22. frac-2neg-rev N/A

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

   3. Applied rewrites 95.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + \frac{x - y \cdot x}{z}} \]

      2. lift-/.f64 N/A

         \[\leadsto y + \color{blue}{\frac{x - y \cdot x}{z}} \]

      3. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

         \[\leadsto \color{blue}{\frac{x - y \cdot x}{z} + \frac{y \cdot z}{z}} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} + \frac{y \cdot z}{z} \]

      7. div-sub N/A

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

      8. mult-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

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

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

      21. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\left(\frac{1}{z} + \frac{y}{x}\right) - \frac{y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\left(\frac{1}{z} + \frac{y}{x}\right) - \frac{y}{z}\right) \]

      6. lower-/.f64 87.8

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

   8. Applied rewrites 87.8%

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

   9. Taylor expanded in z around 0

      \[\leadsto x \cdot \frac{1 - y}{\color{blue}{z}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{1 - y}{z} \]

       2. lower--.f64 58.8

          \[\leadsto x \cdot \frac{1 - y}{z} \]

   11. Applied rewrites 58.8%

       \[\leadsto x \cdot \frac{1 - y}{\color{blue}{z}} \]

   if -2.4000000000000002e66 < x < 3.0000000000000001e38

   1. Initial program 88.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. distribute-lft-in N/A

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

      12. associate--l+ N/A

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

      13. add-to-fraction-rev N/A

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

      14. lower-+.f64 N/A

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

      15. sub-div N/A

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

      16. frac-2neg-rev N/A

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

      17. distribute-lft-neg-out N/A

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

      18. frac-2neg-rev N/A

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

      19. remove-double-neg N/A

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

      20. sub-div N/A

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

      21. sub-negate-rev N/A

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

      22. frac-2neg-rev N/A

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

   3. Applied rewrites 95.9%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + \frac{x - y \cdot x}{z}} \]

      2. lift-/.f64 N/A

         \[\leadsto y + \color{blue}{\frac{x - y \cdot x}{z}} \]

      3. add-to-fraction N/A

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

      4. +-commutative N/A

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

      5. div-add N/A

         \[\leadsto \color{blue}{\frac{x - y \cdot x}{z} + \frac{y \cdot z}{z}} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} + \frac{y \cdot z}{z} \]

      7. div-sub N/A

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

      8. mult-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

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

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. sub-div N/A

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

      20. lower-/.f64 N/A

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

      21. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 78.8%

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

Alternative 8: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+127}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, y, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8e+127) y (if (<= y 1.0) (/ (fma z y x) z) (* (/ y z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8e+127) {
		tmp = y;
	} else if (y <= 1.0) {
		tmp = fma(z, y, x) / z;
	} else {
		tmp = (y / z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8e+127)
		tmp = y;
	elseif (y <= 1.0)
		tmp = Float64(fma(z, y, x) / z);
	else
		tmp = Float64(Float64(y / z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8e+127], y, If[LessEqual[y, 1.0], N[(N[(z * y + x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.79999999999999962e127

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.2%

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

   if -7.79999999999999962e127 < y < 1

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.7%

      \[\leadsto \frac{x + y \cdot \color{blue}{z}}{z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   if 1 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 56.1

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

   6. Applied rewrites 56.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot z \]
   8. Step-by-step derivation

   9. Applied rewrites 37.4%

      \[\leadsto \frac{y}{z} \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{1}{z}, y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (/ 1.0 z) y))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-flip N/A

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

   11. distribute-lft-in N/A

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

   12. associate--l+ N/A

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

   13. add-to-fraction-rev N/A

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

   14. lower-+.f64 N/A

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

   15. sub-div N/A

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

   16. frac-2neg-rev N/A

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

   17. distribute-lft-neg-out N/A

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

   18. frac-2neg-rev N/A

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

   19. remove-double-neg N/A

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

   20. sub-div N/A

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

   21. sub-negate-rev N/A

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

   22. frac-2neg-rev N/A

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

3. Applied rewrites 95.9%

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

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{y + \frac{x - y \cdot x}{z}} \]

   2. lift-/.f64 N/A

      \[\leadsto y + \color{blue}{\frac{x - y \cdot x}{z}} \]

   3. add-to-fraction N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

      \[\leadsto \color{blue}{\frac{x - y \cdot x}{z} + \frac{y \cdot z}{z}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{x - y \cdot x}}{z} + \frac{y \cdot z}{z} \]

   7. div-sub N/A

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

   8. mult-flip N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lift-/.f64 N/A

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

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

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

   14. associate-/l* N/A

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

   15. *-inverses N/A

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

   16. *-rgt-identity N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. sub-div N/A

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

   20. lower-/.f64 N/A

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

   21. lower--.f64 96.1

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 78.8%

   \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{1}}{z}, y\right) \]
9. Add Preprocessing

Alternative 10: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000245:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.000245) y (if (<= y 1.25e-37) (/ x z) (* (/ y z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.000245) {
		tmp = y;
	} else if (y <= 1.25e-37) {
		tmp = x / z;
	} else {
		tmp = (y / z) * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-0.000245d0)) then
        tmp = y
    else if (y <= 1.25d-37) then
        tmp = x / z
    else
        tmp = (y / z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.000245) {
		tmp = y;
	} else if (y <= 1.25e-37) {
		tmp = x / z;
	} else {
		tmp = (y / z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.000245:
		tmp = y
	elif y <= 1.25e-37:
		tmp = x / z
	else:
		tmp = (y / z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.000245)
		tmp = y;
	elseif (y <= 1.25e-37)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(y / z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.000245)
		tmp = y;
	elseif (y <= 1.25e-37)
		tmp = x / z;
	else
		tmp = (y / z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.000245], y, If[LessEqual[y, 1.25e-37], N[(x / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4499999999999999e-4

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.2%

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

   if -2.4499999999999999e-4 < y < 1.2499999999999999e-37

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if 1.2499999999999999e-37 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. lower--.f64 53.6

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - x\right)}{z} \]

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 56.1

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

   6. Applied rewrites 56.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{y}{z} \cdot z \]
   8. Step-by-step derivation

   9. Applied rewrites 37.4%

      \[\leadsto \frac{y}{z} \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000245:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.000245) y (if (<= y 1.25e-37) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.000245) {
		tmp = y;
	} else if (y <= 1.25e-37) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-0.000245d0)) then
        tmp = y
    else if (y <= 1.25d-37) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.000245) {
		tmp = y;
	} else if (y <= 1.25e-37) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.000245:
		tmp = y
	elif y <= 1.25e-37:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.000245)
		tmp = y;
	elseif (y <= 1.25e-37)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.000245)
		tmp = y;
	elseif (y <= 1.25e-37)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.000245], y, If[LessEqual[y, 1.25e-37], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4499999999999999e-4 or 1.2499999999999999e-37 < y

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.2%

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

   if -2.4499999999999999e-4 < y < 1.2499999999999999e-37

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 40.2

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

   4. Applied rewrites 40.2%

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

Alternative 12: 41.2% accurate, 12.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.1%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]

2. Taylor expanded in x around 0

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

4. Applied rewrites 41.2%

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

Reproduce

herbie shell --seed 2025143 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (/ (+ x (* y (- z x))) z))