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

Percentage Accurate: 67.1% → 97.8%

Time: 3.1s

Alternatives: 8

Speedup: 0.6×

• 48.9% 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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 1e+157)
   (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)) <= 1e+157) {
		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)) <= 1e+157)
		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(Float64(x / 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], 1e+157], 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[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.99999999999999983e156

   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. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      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 90.0

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

   3. Applied rewrites 90.0%

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

   if 9.99999999999999983e156 < (/.f64 (*.f64 z z) (*.f64 t t))

   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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 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 89.5

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

   3. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 2: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e+252)
   (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)) <= 2e+252) {
		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)) <= 2e+252)
		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], 2e+252], 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)) < 2.0000000000000002e252

   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. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      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 90.0

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

   3. Applied rewrites 90.0%

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

   if 2.0000000000000002e252 < (/.f64 (*.f64 z z) (*.f64 t t))

   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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 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 89.5

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

   3. Applied rewrites 89.5%

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

Alternative 3: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{-217}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e-217)
   (* (/ z t) (/ z t))
   (fma (/ x y) (/ x y) (* (/ z (* t t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e-217) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = fma((x / y), (x / y), ((z / (t * t)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e-217)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z / Float64(t * t)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e-217], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.00000000000000008e-217

   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 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 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 59.1

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

   6. Applied rewrites 59.1%

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

   if 1.00000000000000008e-217 < (/.f64 (*.f64 x x) (*.f64 y y))

   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. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      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 90.0

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

   3. Applied rewrites 90.0%

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

Alternative 4: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{if}\;t\_1 \leq 10^{-217}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z t) (/ z t))))
   (if (<= t_1 1e-217)
     t_2
     (if (<= t_1 INFINITY) (fma (/ z (* t t)) z (* (/ 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) * (z / t);
	double tmp;
	if (t_1 <= 1e-217) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((z / (t * t)), z, ((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 / t) * Float64(z / t))
	tmp = 0.0
	if (t_1 <= 1e-217)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / Float64(y * y)) * x));
	else
		tmp = t_2;
	end
	return 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 / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-217], t$95$2, 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], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.00000000000000008e-217 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   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 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 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 59.1

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

   6. Applied rewrites 59.1%

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

   if 1.00000000000000008e-217 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 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 81.0

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

   3. Applied rewrites 81.0%

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

Alternative 5: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+186}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{t}{y} \cdot x\right) \cdot x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z t) (/ z t))))
   (if (<= t_1 2e+186)
     t_2
     (if (<= t_1 INFINITY) (/ (* (* (/ t y) x) x) (* t y)) 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) * (z / t);
	double tmp;
	if (t_1 <= 2e+186) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((t / y) * x) * x) / (t * y);
	} 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) * (z / t);
	double tmp;
	if (t_1 <= 2e+186) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((t / y) * x) * x) / (t * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / t) * (z / t)
	tmp = 0
	if t_1 <= 2e+186:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (((t / y) * x) * x) / (t * y)
	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 / t) * Float64(z / t))
	tmp = 0.0
	if (t_1 <= 2e+186)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(t / y) * x) * x) / Float64(t * y));
	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) * (z / t);
	tmp = 0.0;
	if (t_1 <= 2e+186)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (((t / y) * x) * x) / (t * y);
	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 / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+186], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(t / y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999996e186 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   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 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 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 59.1

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

   6. Applied rewrites 59.1%

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

   if 1.99999999999999996e186 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. frac-add N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 81.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-/.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. pow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

Alternative 6: 59.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+262}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z \cdot z}{-t}}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.4e+262) (* (/ z t) (/ z t)) (/ (/ (* z z) (- t)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.4e+262) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = ((z * z) / -t) / -t;
	}
	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, 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
    real(8) :: tmp
    if (x <= 2.4d+262) then
        tmp = (z / t) * (z / t)
    else
        tmp = ((z * z) / -t) / -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.4e+262) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = ((z * z) / -t) / -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.4e+262:
		tmp = (z / t) * (z / t)
	else:
		tmp = ((z * z) / -t) / -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.4e+262)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(Float64(z * z) / Float64(-t)) / Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.4e+262)
		tmp = (z / t) * (z / t);
	else
		tmp = ((z * z) / -t) / -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.4e+262], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] / (-t)), $MachinePrecision] / (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999983e262

   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 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 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 59.1

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

   6. Applied rewrites 59.1%

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

   if 2.39999999999999983e262 < x

   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 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 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l/ N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 49.2

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

   6. Applied rewrites 49.2%

      \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. sqr-neg-rev N/A

         \[\leadsto \frac{{z}^{2}}{\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{{z}^{2}}{\mathsf{neg}\left(t\right)}}{\color{blue}{\mathsf{neg}\left(t\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{z}^{2}}{\mathsf{neg}\left(t\right)}}{\color{blue}{\mathsf{neg}\left(t\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{{z}^{2}}{\mathsf{neg}\left(t\right)}}{\mathsf{neg}\left(\color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{z \cdot z}{\mathsf{neg}\left(t\right)}}{\mathsf{neg}\left(t\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{z \cdot z}{\mathsf{neg}\left(t\right)}}{\mathsf{neg}\left(t\right)} \]

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 7: 59.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $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 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 53.2

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

4. Applied rewrites 53.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 59.1

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

6. Applied rewrites 59.1%

   \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]
7. Add Preprocessing

Alternative 8: 53.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $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 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 53.2

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

4. Applied rewrites 53.2%

   \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))