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

Percentage Accurate: 67.3% → 98.4%

Time: 5.3s

Alternatives: 7

Speedup: 1.0×

• 48.8% 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: 67.3% 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: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y\_m} \cdot x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y\_m}}{y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 1e-191)
   (fma (/ z t) (/ z t) (/ (* (/ x y_m) x) y_m))
   (fma (/ z t) (/ z t) (* (/ (/ x y_m) y_m) x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 1e-191) {
		tmp = fma((z / t), (z / t), (((x / y_m) * x) / y_m));
	} else {
		tmp = fma((z / t), (z / t), (((x / y_m) / y_m) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 1e-191)
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(Float64(x / y_m) * x) / y_m));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(Float64(x / y_m) / y_m) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 1e-191], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e-191

   1. Initial program 67.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 90.0

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

   3. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 1e-191 < y

   1. Initial program 67.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 90.0

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

   3. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)} \]
   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. lower-/.f64 N/A

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

      5. lift-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 2: 96.8% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (fma (/ z t) (/ z t) (* (/ (/ x y_m) y_m) x)))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 90.0

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

3. Applied rewrites 90.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)} \]
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. lower-/.f64 N/A

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

   5. lift-/.f64 96.8

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

5. Applied rewrites 96.8%

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

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, \frac{x}{y\_m}, \frac{z}{t \cdot t} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y\_m \cdot y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+299)
   (fma (/ x y_m) (/ x y_m) (* (/ z (* t t)) z))
   (fma (/ z t) (/ z t) (* (/ x (* y_m y_m)) x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+299) {
		tmp = fma((x / y_m), (x / y_m), ((z / (t * t)) * z));
	} else {
		tmp = fma((z / t), (z / t), ((x / (y_m * y_m)) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+299)
		tmp = fma(Float64(x / y_m), Float64(x / y_m), Float64(Float64(z / Float64(t * t)) * z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x / Float64(y_m * y_m)) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e+299], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $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$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000003e299

   1. Initial program 67.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 89.5

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

   3. Applied rewrites 89.5%

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

   if 5.0000000000000003e299 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 67.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 90.0

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

   3. Applied rewrites 90.0%

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

Alternative 4: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y\_m \cdot y\_m} \leq 0:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, \frac{x}{y\_m}, \frac{z}{t \cdot t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* x x) (* y_m y_m)) 0.0)
   (* (/ z t) (/ z t))
   (fma (/ x y_m) (/ x y_m) (* (/ z (* t t)) z))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 0.0) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = fma((x / y_m), (x / y_m), ((z / (t * t)) * z));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y_m * y_m)) <= 0.0)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = fma(Float64(x / y_m), Float64(x / y_m), Float64(Float64(z / Float64(t * t)) * z));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $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)) < 0.0

   1. Initial program 67.3%

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

   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 52.7

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

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]
   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-*l/ N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. 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 0.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 67.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 89.5

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

   3. Applied rewrites 89.5%

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

Alternative 5: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 1.25 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y\_m \cdot y\_m} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (* y_m y_m) 1.25e+286)
   (fma (/ z (* t t)) z (* (/ x (* y_m y_m)) x))
   (* (/ z t) (/ z t))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if ((y_m * y_m) <= 1.25e+286) {
		tmp = fma((z / (t * t)), z, ((x / (y_m * y_m)) * x));
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 1.25e+286)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / Float64(y_m * y_m)) * x));
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 1.25e+286], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.2500000000000001e286

   1. Initial program 67.3%

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

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

   if 1.2500000000000001e286 < (*.f64 y y)

   1. Initial program 67.3%

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

   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 52.7

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

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]
   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-*l/ N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 59.1

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

   6. Applied rewrites 59.1%

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

Alternative 6: 59.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{z}{t} \cdot \frac{z}{t} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (* (/ z t) (/ z t)))

C

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

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return (z / t) * (z / t);
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return (z / t) * (z / t)

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

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 52.7

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

4. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]
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-*l/ N/A

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

   5. frac-times N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. 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 7: 52.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{z}{t \cdot t} \cdot z \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (* (/ z (* t t)) z))

C

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

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return (z / (t * t)) * z;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return (z / (t * t)) * z

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

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 52.7

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

4. Applied rewrites 52.7%

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

Reproduce

herbie shell --seed 2025142 
(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))))