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

Percentage Accurate: 66.5% → 97.7%

Time: 4.3s

Alternatives: 9

Speedup: 0.9×

• 47.6% 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: 66.5% 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: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{-250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y\_m} \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y\_m}}{y\_m}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 8e-250)
   (fma (/ z (* t t)) z (* (/ x y_m) (/ x y_m)))
   (fma (/ (/ 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 <= 8e-250) {
		tmp = fma((z / (t * t)), z, ((x / y_m) * (x / y_m)));
	} else {
		tmp = fma(((x / y_m) / y_m), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 8e-250)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / y_m) * Float64(x / y_m)));
	else
		tmp = fma(Float64(Float64(x / y_m) / y_m), x, 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[y$95$m, 8e-250], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.0000000000000004e-250

   1. Initial program 64.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 8.0000000000000004e-250 < y

   1. Initial program 64.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. pow2 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. pow2 N/A

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

      20. times-frac N/A

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

      21. pow2 N/A

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

      22. lower-pow.f64 N/A

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

      23. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 97.3

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

   6. Applied rewrites 97.3%

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

Alternative 2: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-316} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y\_m} \cdot x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m \cdot y\_m} \cdot x + t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (or (<= t_1 5e-316) (not (<= t_1 INFINITY)))
     (/ (* (/ x y_m) x) y_m)
     (+ (* (/ x (* y_m y_m)) x) t_1))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 5e-316) || !(t_1 <= ((double) INFINITY))) {
		tmp = ((x / y_m) * x) / y_m;
	} else {
		tmp = ((x / (y_m * y_m)) * x) + t_1;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 5e-316) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = ((x / y_m) * x) / y_m;
	} else {
		tmp = ((x / (y_m * y_m)) * x) + t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if (t_1 <= 5e-316) or not (t_1 <= math.inf):
		tmp = ((x / y_m) * x) / y_m
	else:
		tmp = ((x / (y_m * y_m)) * x) + t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if ((t_1 <= 5e-316) || !(t_1 <= Inf))
		tmp = Float64(Float64(Float64(x / y_m) * x) / y_m);
	else
		tmp = Float64(Float64(Float64(x / Float64(y_m * y_m)) * x) + t_1);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if ((t_1 <= 5e-316) || ~((t_1 <= Inf)))
		tmp = ((x / y_m) * x) / y_m;
	else
		tmp = ((x / (y_m * y_m)) * x) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-316], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.000000017e-316 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 50.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 55.0

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

   7. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 55.5

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

   9. Applied rewrites 55.5%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 74.3

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

   12. Applied rewrites 74.3%

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

   if 5.000000017e-316 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 86.2

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

   6. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \cdot x + \frac{z \cdot z}{t \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-316} \lor \neg \left(\frac{z \cdot z}{t \cdot t} \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + \frac{z \cdot z}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-108} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y\_m} \cdot x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (or (<= t_1 5e-108) (not (<= t_1 INFINITY)))
     (/ (* (/ x y_m) x) y_m)
     t_1)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 5e-108) || !(t_1 <= ((double) INFINITY))) {
		tmp = ((x / y_m) * x) / y_m;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 5e-108) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = ((x / y_m) * x) / y_m;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if (t_1 <= 5e-108) or not (t_1 <= math.inf):
		tmp = ((x / y_m) * x) / y_m
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if ((t_1 <= 5e-108) || !(t_1 <= Inf))
		tmp = Float64(Float64(Float64(x / y_m) * x) / y_m);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if ((t_1 <= 5e-108) || ~((t_1 <= Inf)))
		tmp = ((x / y_m) * x) / y_m;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e-108], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5e-108 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 52.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 54.2

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

   9. Applied rewrites 54.2%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 73.0

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

   12. Applied rewrites 73.0%

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

   if 5e-108 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 96.1

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

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 96.1

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

   6. Applied rewrites 96.1%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 88.7%

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

   10. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 78.1

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

   12. Applied rewrites 78.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-108} \lor \neg \left(\frac{z \cdot z}{t \cdot t} \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot z}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y\_m \cdot y\_m}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+34}:\\ \;\;\;\;t\_1 + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y\_m} \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y_m y_m))))
   (if (<= t_1 5e+34)
     (+ t_1 (* (/ z t) (/ z t)))
     (fma (/ z (* t t)) z (* (/ x y_m) (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (x * x) / (y_m * y_m);
	double tmp;
	if (t_1 <= 5e+34) {
		tmp = t_1 + ((z / t) * (z / t));
	} else {
		tmp = fma((z / (t * t)), z, ((x / y_m) * (x / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y_m * y_m))
	tmp = 0.0
	if (t_1 <= 5e+34)
		tmp = Float64(t_1 + Float64(Float64(z / t) * Float64(z / t)));
	else
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / y_m) * Float64(x / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+34], N[(t$95$1 + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 4.9999999999999998e34

   1. Initial program 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   if 4.9999999999999998e34 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 60.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

Alternative 5: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y\_m \cdot y\_m}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y\_m} \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y_m y_m))))
   (if (<= t_1 5e-46)
     (fma (/ (/ z t) t) z t_1)
     (fma (/ z (* t t)) z (* (/ x y_m) (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (x * x) / (y_m * y_m);
	double tmp;
	if (t_1 <= 5e-46) {
		tmp = fma(((z / t) / t), z, t_1);
	} else {
		tmp = fma((z / (t * t)), z, ((x / y_m) * (x / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y_m * y_m))
	tmp = 0.0
	if (t_1 <= 5e-46)
		tmp = fma(Float64(Float64(z / t) / t), z, t_1);
	else
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / y_m) * Float64(x / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-46], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + t$95$1), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 4.99999999999999992e-46

   1. Initial program 66.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 96.1

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

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 90.4

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

   6. Applied rewrites 90.4%

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

   if 4.99999999999999992e-46 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 98.5

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

   6. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 95.3

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

   8. Applied rewrites 95.3%

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

Alternative 6: 96.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. pow2 N/A

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

   13. pow2 N/A

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

   14. lower-fma.f64 N/A

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

   15. pow2 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. pow2 N/A

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

   20. pow2 N/A

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

   21. times-frac N/A

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

   22. pow2 N/A

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

   23. lower-pow.f64 N/A

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

   24. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 97.5

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

6. Applied rewrites 97.5%

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

Alternative 7: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \left|x\right| \cdot \frac{\left|x\right|}{y\_m \cdot y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y\_m} \cdot x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= t 4.8e+115)
   (fma (/ z (* t t)) z (* (fabs x) (/ (fabs x) (* y_m y_m))))
   (/ (* (/ x y_m) x) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (t <= 4.8e+115) {
		tmp = fma((z / (t * t)), z, (fabs(x) * (fabs(x) / (y_m * y_m))));
	} else {
		tmp = ((x / y_m) * x) / y_m;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (t <= 4.8e+115)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(abs(x) * Float64(abs(x) / Float64(y_m * y_m))));
	else
		tmp = Float64(Float64(Float64(x / y_m) * x) / y_m);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[t, 4.8e+115], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000001e115

   1. Initial program 66.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. times-frac N/A

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

      22. pow2 N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. sqr-abs-rev N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 86.5

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

   6. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.1

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

   8. Applied rewrites 79.1%

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

   if 4.8000000000000001e115 < t

   1. Initial program 51.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. frac-add N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 59.3

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

   7. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 59.8

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

   9. Applied rewrites 59.8%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 83.5

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

   12. Applied rewrites 83.5%

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

Alternative 8: 89.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. pow2 N/A

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

   13. pow2 N/A

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

   14. lower-fma.f64 N/A

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

   15. pow2 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. pow2 N/A

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

   20. pow2 N/A

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

   21. times-frac N/A

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

   22. pow2 N/A

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

   23. lower-pow.f64 N/A

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

   24. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 97.5

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

6. Applied rewrites 97.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. pow2 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 88.8

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

8. Applied rewrites 88.8%

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

Alternative 9: 47.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. pow2 N/A

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

   13. pow2 N/A

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

   14. lower-fma.f64 N/A

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

   15. pow2 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. pow2 N/A

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

   20. pow2 N/A

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

   21. times-frac N/A

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

   22. pow2 N/A

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

   23. lower-pow.f64 N/A

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

   24. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, {\left(\frac{x}{y}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 97.5

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 64.1%

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

10. Taylor expanded in x around 0

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

    1. pow2 N/A

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

    2. lift-*.f64 45.4

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

12. Applied rewrites 45.4%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (pow (/ x y) 2) (pow (/ z t) 2)))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))