Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Percentage Accurate: 67.1% → 96.9%

Time: 2.9s

Alternatives: 8

Speedup: 1.0×

• 47.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

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

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

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

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ x y) (/ x y) (* (/ (/ z t) t) z)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. times-frac N/A

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

   9. pow2 N/A

      \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   10. pow2 N/A

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{{z}^{2}}{{t}^{2}}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{{z}^{2}}{{t}^{2}}\right) \]

   14. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{{t}^{2}}\right) \]

   15. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{z \cdot \frac{z}{{t}^{2}}}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}} \cdot z}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}} \cdot z}\right) \]

   18. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}}} \cdot z\right) \]

   19. pow2 N/A

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

   20. lift-*.f64 89.8

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

3. Applied rewrites 89.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 96.9

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

5. Applied rewrites 96.9%

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

Alternative 2: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+263)
   (fma (/ x y) (/ x y) (* (/ z (* t t)) z))
   (fma (/ z t) (/ z t) (* (/ x (* y y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+263) {
		tmp = fma((x / y), (x / y), ((z / (t * t)) * z));
	} else {
		tmp = fma((z / t), (z / t), ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+263)
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z / Float64(t * t)) * z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e+263], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000022e263

   1. Initial program 73.7%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. times-frac N/A

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

      9. pow2 N/A

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{{z}^{2}}{{t}^{2}}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{{z}^{2}}{{t}^{2}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{{z}^{2}}{{t}^{2}}\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{{t}^{2}}\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{z \cdot \frac{z}{{t}^{2}}}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}} \cdot z}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}} \cdot z}\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}}} \cdot z\right) \]

      19. pow2 N/A

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

      20. lift-*.f64 97.3

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

   3. Applied rewrites 97.3%

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

   if 5.00000000000000022e263 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 59.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

          \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \frac{\color{blue}{{x}^{2}}}{y \cdot y} \]

      11. pow2 N/A

          \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \frac{{x}^{2}}{\color{blue}{{y}^{2}}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{{x}^{2}}{{y}^{2}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, \frac{z}{t}, \frac{{x}^{2}}{{y}^{2}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{\frac{z}{t}}, \frac{{x}^{2}}{{y}^{2}}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{{y}^{2}}\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{x \cdot \frac{x}{{y}^{2}}}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{{y}^{2}} \cdot x}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{{y}^{2}} \cdot x}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{{y}^{2}}} \cdot x\right) \]

      20. pow2 N/A

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

      21. lift-*.f64 95.4

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

   3. Applied rewrites 95.4%

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

Alternative 3: 89.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ x y) (/ x y) (* (/ z (* t t)) z)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. times-frac N/A

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

   9. pow2 N/A

      \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{\color{blue}{{z}^{2}}}{t \cdot t} \]

   10. pow2 N/A

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{{z}^{2}}{\color{blue}{{t}^{2}}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{{z}^{2}}{{t}^{2}}\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{{z}^{2}}{{t}^{2}}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{{z}^{2}}{{t}^{2}}\right) \]

   14. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{{t}^{2}}\right) \]

   15. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{z \cdot \frac{z}{{t}^{2}}}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}} \cdot z}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}} \cdot z}\right) \]

   18. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z}{{t}^{2}}} \cdot z\right) \]

   19. pow2 N/A

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

   20. lift-*.f64 89.8

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

3. Applied rewrites 89.8%

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

Alternative 4: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e-286)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY)
       (fma (/ z (* t t)) z (* (/ x (* y y)) x))
       (/ (* (/ x y) x) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e-286) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((z / (t * t)), z, ((x / (y * y)) * x));
	} else {
		tmp = ((x / y) * x) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e-286)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / Float64(y * y)) * x));
	else
		tmp = Float64(Float64(Float64(x / y) * x) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-286], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000037e-286

   1. Initial program 70.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

      6. pow2 N/A

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

      7. lift-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 93.4

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

   6. Applied rewrites 93.4%

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

   if 5.00000000000000037e-286 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 79.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{{t}^{2}}} + \frac{x \cdot x}{y \cdot y} \]

      10. associate-/l* N/A

          \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} + \frac{x \cdot x}{y \cdot y} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} + \frac{x \cdot x}{y \cdot y} \]

      12. pow2 N/A

          \[\leadsto \frac{z}{{t}^{2}} \cdot z + \frac{\color{blue}{{x}^{2}}}{y \cdot y} \]

      13. pow2 N/A

          \[\leadsto \frac{z}{{t}^{2}} \cdot z + \frac{{x}^{2}}{\color{blue}{{y}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{{t}^{2}}, z, \frac{{x}^{2}}{{y}^{2}}\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{{t}^{2}}}, z, \frac{{x}^{2}}{{y}^{2}}\right) \]

      16. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{t \cdot t}}, z, \frac{{x}^{2}}{{y}^{2}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{t \cdot t}}, z, \frac{{x}^{2}}{{y}^{2}}\right) \]

      18. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\color{blue}{x \cdot x}}{{y}^{2}}\right) \]

      19. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \color{blue}{x \cdot \frac{x}{{y}^{2}}}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \color{blue}{\frac{x}{{y}^{2}} \cdot x}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \color{blue}{\frac{x}{{y}^{2}} \cdot x}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \color{blue}{\frac{x}{{y}^{2}}} \cdot x\right) \]

      23. pow2 N/A

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

      24. lift-*.f64 91.3

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

   3. Applied rewrites 91.3%

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

   if +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

      6. pow2 N/A

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

      7. lift-*.f64 40.9

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

   4. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 46.4

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

   6. Applied rewrites 46.4%

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

Alternative 5: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-81)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY) (* (/ z (* t t)) z) (/ (* (/ x y) x) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-81) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = ((x / y) * x) / y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-81) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = ((x / y) * x) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-81:
		tmp = (x / y) * (x / y)
	elif t_1 <= math.inf:
		tmp = (z / (t * t)) * z
	else:
		tmp = ((x / y) * x) / y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-81)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	else
		tmp = Float64(Float64(Float64(x / y) * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 2e-81)
		tmp = (x / y) * (x / y);
	elseif (t_1 <= Inf)
		tmp = (z / (t * t)) * z;
	else
		tmp = ((x / y) * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-81], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999999e-81

   1. Initial program 71.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

      6. pow2 N/A

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

      7. lift-*.f64 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 89.1

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

   6. Applied rewrites 89.1%

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

   if 1.9999999999999999e-81 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{z \cdot z}{{\color{blue}{t}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot z \]

      6. pow2 N/A

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

      7. lift-*.f64 82.1

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

   4. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

   if +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

      6. pow2 N/A

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

      7. lift-*.f64 40.9

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

   4. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 46.4

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

   6. Applied rewrites 46.4%

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

Alternative 6: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* (/ x y) (/ x y))))
   (if (<= t_1 2e-81) t_2 (if (<= t_1 INFINITY) (* (/ z (* t t)) z) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = (x / y) * (x / y);
	double tmp;
	if (t_1 <= 2e-81) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = (x / y) * (x / y);
	double tmp;
	if (t_1 <= 2e-81) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = (x / y) * (x / y)
	tmp = 0
	if t_1 <= 2e-81:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (z / (t * t)) * z
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(Float64(x / y) * Float64(x / y))
	tmp = 0.0
	if (t_1 <= 2e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = (x / y) * (x / y);
	tmp = 0.0;
	if (t_1 <= 2e-81)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (z / (t * t)) * z;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-81], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999999e-81 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

      6. pow2 N/A

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

      7. lift-*.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 80.7

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

   6. Applied rewrites 80.7%

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

   if 1.9999999999999999e-81 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 78.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{z \cdot z}{{\color{blue}{t}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot z \]

      6. pow2 N/A

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

      7. lift-*.f64 82.1

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

   4. Applied rewrites 82.1%

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

Alternative 7: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t \cdot t} \cdot z\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-189}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z (* t t)) z)))
   (if (<= t_1 4e-189) t_2 (if (<= t_1 INFINITY) (* (/ x (* y y)) x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / (t * t)) * z;
	double tmp;
	if (t_1 <= 4e-189) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / (t * t)) * z;
	double tmp;
	if (t_1 <= 4e-189) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / (t * t)) * z
	tmp = 0
	if t_1 <= 4e-189:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / (y * y)) * x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(z / Float64(t * t)) * z)
	tmp = 0.0
	if (t_1 <= 4e-189)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / Float64(y * y)) * x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = (z / (t * t)) * z;
	tmp = 0.0;
	if (t_1 <= 4e-189)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / (y * y)) * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-189], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 4.00000000000000027e-189 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 53.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{z \cdot z}{{\color{blue}{t}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot \color{blue}{z} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z}{{t}^{2}} \cdot z \]

      6. pow2 N/A

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

      7. lift-*.f64 66.2

         \[\leadsto \frac{z}{t \cdot t} \cdot z \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

   if 4.00000000000000027e-189 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 80.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

      6. pow2 N/A

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

      7. lift-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

Alternative 8: 53.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ x (* y y)) x))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x / (y * y)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
3. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{x \cdot x}{{\color{blue}{y}}^{2}} \]

   2. associate-/l* N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{{y}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot \color{blue}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{x}{{y}^{2}} \cdot x \]

   6. pow2 N/A

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

   7. lift-*.f64 53.1

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

4. Applied rewrites 53.1%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64
  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))