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

Percentage Accurate: 67.5% → 98.7%

Time: 5.9s

Alternatives: 5

Speedup: 1.0×

• 48.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% 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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e+234)
   (fma (/ (/ z t) t) z (* (/ x y) (/ x y)))
   (fma (/ z t) (/ z t) (* (/ (/ x y) y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e+234) {
		tmp = fma(((z / t) / t), z, ((x / y) * (x / y)));
	} else {
		tmp = fma((z / t), (z / t), (((x / y) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e+234)
		tmp = fma(Float64(Float64(z / t) / t), z, Float64(Float64(x / y) * Float64(x / y)));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e+234], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.00000000000000004e234

   1. Initial program 67.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. sub-flip-reverse N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

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

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

      11. remove-double-neg N/A

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

      12. mult-flip N/A

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

      13. lift-/.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 74.3

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 81.1

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

   3. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

   if 2.00000000000000004e234 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 67.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. sub-flip-reverse N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

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

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

      12. remove-double-neg N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 81.9

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l/ N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 89.4

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

   3. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 2: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 5e+298)
   (fma (/ z t) (/ z t) (* (/ x (* y y)) x))
   (fma (/ x y) (/ x y) (* (/ z (* t t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5e+298) {
		tmp = fma((z / t), (z / t), ((x / (y * y)) * x));
	} else {
		tmp = fma((x / y), (x / y), ((z / (t * t)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 5e+298)
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x / Float64(y * y)) * x));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z / Float64(t * t)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+298], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5.0000000000000003e298

   1. Initial program 67.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. sub-flip-reverse N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

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

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

      12. remove-double-neg N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 81.9

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l/ N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 89.4

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

   3. Applied rewrites 89.4%

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

   if 5.0000000000000003e298 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 67.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

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

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

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

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

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

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

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

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

      15. associate-/l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 90.1%

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

Alternative 3: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. sub-flip-reverse N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

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

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

   12. remove-double-neg N/A

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

   13. mult-flip N/A

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

   14. lift-/.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 81.9

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

   18. lift-/.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*l/ N/A

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

   21. lower-*.f64 N/A

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

   22. lower-/.f64 89.4

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

3. Applied rewrites 89.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 96.7

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

5. Applied rewrites 96.7%

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

Alternative 4: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

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

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

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

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

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

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

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

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

   15. associate-/l* N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. remove-double-neg N/A

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

   19. lower-fma.f64 N/A

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

3. Applied rewrites 90.1%

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

Alternative 5: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. sub-flip-reverse N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

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

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

   11. remove-double-neg N/A

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

   12. mult-flip N/A

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

   13. lift-/.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-/.f64 74.3

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*l/ N/A

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

   19. lower-*.f64 N/A

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

   20. lower-/.f64 81.1

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

3. Applied rewrites 81.1%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64
  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))