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

Percentage Accurate: 65.8% → 96.5%

Time: 3.7s

Alternatives: 10

Speedup: 1.0×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t}, \frac{z\_m}{t}, \frac{\frac{x}{y} \cdot x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t}, \frac{z\_m}{t}, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3e-157)
   (fma (/ z_m t) (/ z_m t) (/ (* (/ x y) x) y))
   (fma (/ z_m t) (/ z_m t) (* (/ (/ x y) y) x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3e-157) {
		tmp = fma((z_m / t), (z_m / t), (((x / y) * x) / y));
	} else {
		tmp = fma((z_m / t), (z_m / t), (((x / y) / y) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3e-157)
		tmp = fma(Float64(z_m / t), Float64(z_m / t), Float64(Float64(Float64(x / y) * x) / y));
	else
		tmp = fma(Float64(z_m / t), Float64(z_m / t), Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3e-157], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3e-157

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 89.2

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

   3. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if 3e-157 < z

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 89.2

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

   3. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 2: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t}, \frac{z\_m}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z\_m}{t \cdot t} \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e+196)
   (fma (/ z_m t) (/ z_m t) (* (/ x (* y y)) x))
   (fma (/ x y) (/ x y) (* (/ z_m (* t t)) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+196) {
		tmp = fma((z_m / t), (z_m / t), ((x / (y * y)) * x));
	} else {
		tmp = fma((x / y), (x / y), ((z_m / (t * t)) * z_m));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e+196)
		tmp = fma(Float64(z_m / t), Float64(z_m / t), Float64(Float64(x / Float64(y * y)) * x));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z_m / Float64(t * t)) * z_m));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e+196], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / 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$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999999999999995e195

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 89.2

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

   3. Applied rewrites 89.2%

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

   if 9.9999999999999995e195 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 88.8

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

   3. Applied rewrites 88.8%

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

Alternative 3: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. times-frac N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. associate-/l* N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. pow2 N/A

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

   21. lift-*.f64 89.2

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

3. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. lower-/.f64 96.5

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

5. Applied rewrites 96.5%

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

Alternative 4: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{-224}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \left(\frac{1}{t} \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z\_m}{t \cdot t} \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e-224)
   (* (/ z_m t) (* (/ 1.0 t) z_m))
   (fma (/ x y) (/ x y) (* (/ z_m (* t t)) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e-224) {
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	} else {
		tmp = fma((x / y), (x / y), ((z_m / (t * t)) * z_m));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e-224)
		tmp = Float64(Float64(z_m / t) * Float64(Float64(1.0 / t) * z_m));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z_m / Float64(t * t)) * z_m));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e-224], N[(N[(z$95$m / t), $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1e-224

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 59.1

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

   6. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. frac-times N/A

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

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

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

      8. pow2 N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot {z}^{2}}{\color{blue}{t} \cdot \left(-t\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{{z}^{2} \cdot -1}{\color{blue}{t} \cdot \left(-t\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{{z}^{2}}{t} \cdot \color{blue}{\frac{-1}{-t}} \]

      12. pow2 N/A

          \[\leadsto \frac{z \cdot z}{t} \cdot \frac{-1}{-t} \]

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{z}{t} \cdot z\right) \cdot \frac{-1}{-t} \]

      15. lift-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 59.1

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

      20. lift-/.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{-1}{-t} \cdot z\right) \]

      21. metadata-eval N/A

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

      22. lift-neg.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(t\right)} \cdot z\right) \]

      23. frac-2neg N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

      24. lower-/.f64 59.1

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

   8. Applied rewrites 59.1%

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

   if 1e-224 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 88.8

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

   3. Applied rewrites 88.8%

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

Alternative 5: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{-224}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \left(\frac{1}{t} \cdot z\_m\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t \cdot t}, z\_m, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z\_m}{t}}{t} \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 1e-224)
     (* (/ z_m t) (* (/ 1.0 t) z_m))
     (if (<= t_1 INFINITY)
       (fma (/ z_m (* t t)) z_m (* (/ x (* y y)) x))
       (* (/ (/ z_m t) t) z_m)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e-224) {
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((z_m / (t * t)), z_m, ((x / (y * y)) * x));
	} else {
		tmp = ((z_m / t) / t) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 1e-224)
		tmp = Float64(Float64(z_m / t) * Float64(Float64(1.0 / t) * z_m));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(z_m / Float64(t * t)), z_m, Float64(Float64(x / Float64(y * y)) * x));
	else
		tmp = Float64(Float64(Float64(z_m / t) / t) * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-224], N[(N[(z$95$m / t), $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1e-224

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 59.1

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

   6. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. frac-times N/A

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

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

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

      8. pow2 N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot {z}^{2}}{\color{blue}{t} \cdot \left(-t\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{{z}^{2} \cdot -1}{\color{blue}{t} \cdot \left(-t\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{{z}^{2}}{t} \cdot \color{blue}{\frac{-1}{-t}} \]

      12. pow2 N/A

          \[\leadsto \frac{z \cdot z}{t} \cdot \frac{-1}{-t} \]

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{z}{t} \cdot z\right) \cdot \frac{-1}{-t} \]

      15. lift-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 59.1

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

      20. lift-/.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{-1}{-t} \cdot z\right) \]

      21. metadata-eval N/A

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

      22. lift-neg.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(t\right)} \cdot z\right) \]

      23. frac-2neg N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

      24. lower-/.f64 59.1

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

   8. Applied rewrites 59.1%

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

   if 1e-224 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. pow2 N/A

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

      19. associate-/l* N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. pow2 N/A

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

      24. lift-*.f64 79.6

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

   3. Applied rewrites 79.6%

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

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

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 57.0

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

   6. Applied rewrites 57.0%

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

Alternative 6: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{-38}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \left(\frac{1}{t} \cdot z\_m\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z\_m}{t}}{t} \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 1e-38)
     (* (/ z_m t) (* (/ 1.0 t) z_m))
     (if (<= t_1 INFINITY)
       (/ (* (* (/ x y) x) t) (* t y))
       (* (/ (/ z_m t) t) z_m)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e-38) {
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z_m / t) / t) * z_m;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e-38) {
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z_m / t) / t) * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if t_1 <= 1e-38:
		tmp = (z_m / t) * ((1.0 / t) * z_m)
	elif t_1 <= math.inf:
		tmp = (((x / y) * x) * t) / (t * y)
	else:
		tmp = ((z_m / t) / t) * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 1e-38)
		tmp = Float64(Float64(z_m / t) * Float64(Float64(1.0 / t) * z_m));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) * t) / Float64(t * y));
	else
		tmp = Float64(Float64(Float64(z_m / t) / t) * z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (x * x) / (y * y);
	tmp = 0.0;
	if (t_1 <= 1e-38)
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	elseif (t_1 <= Inf)
		tmp = (((x / y) * x) * t) / (t * y);
	else
		tmp = ((z_m / t) / t) * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-38], N[(N[(z$95$m / t), $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999999999999996e-39

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 59.1

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

   6. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. frac-times N/A

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

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

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

      8. pow2 N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot {z}^{2}}{\color{blue}{t} \cdot \left(-t\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{{z}^{2} \cdot -1}{\color{blue}{t} \cdot \left(-t\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{{z}^{2}}{t} \cdot \color{blue}{\frac{-1}{-t}} \]

      12. pow2 N/A

          \[\leadsto \frac{z \cdot z}{t} \cdot \frac{-1}{-t} \]

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{z}{t} \cdot z\right) \cdot \frac{-1}{-t} \]

      15. lift-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 59.1

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

      20. lift-/.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{-1}{-t} \cdot z\right) \]

      21. metadata-eval N/A

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

      22. lift-neg.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(t\right)} \cdot z\right) \]

      23. frac-2neg N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

      24. lower-/.f64 59.1

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

   8. Applied rewrites 59.1%

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

   if 9.9999999999999996e-39 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 65.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. 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

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

   3. Applied rewrites 81.2%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 47.7

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

   6. Applied rewrites 47.7%

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

   7. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 51.6

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

   9. Applied rewrites 51.6%

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

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

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 57.0

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

   6. Applied rewrites 57.0%

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

Alternative 7: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{z\_m \cdot z\_m}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \left(\frac{1}{t} \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= y 6e-222) (/ (/ (* z_m z_m) t) t) (* (/ z_m t) (* (/ 1.0 t) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (y <= 6e-222) {
		tmp = ((z_m * z_m) / t) / t;
	} else {
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	}
	return tmp;
}

Fortran

z_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, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 6d-222) then
        tmp = ((z_m * z_m) / t) / t
    else
        tmp = (z_m / t) * ((1.0d0 / t) * z_m)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (y <= 6e-222) {
		tmp = ((z_m * z_m) / t) / t;
	} else {
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if y <= 6e-222:
		tmp = ((z_m * z_m) / t) / t
	else:
		tmp = (z_m / t) * ((1.0 / t) * z_m)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (y <= 6e-222)
		tmp = Float64(Float64(Float64(z_m * z_m) / t) / t);
	else
		tmp = Float64(Float64(z_m / t) * Float64(Float64(1.0 / t) * z_m));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (y <= 6e-222)
		tmp = ((z_m * z_m) / t) / t;
	else
		tmp = (z_m / t) * ((1.0 / t) * z_m);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[y, 6e-222], N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000059e-222

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l/ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 6.00000000000000059e-222 < y

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 59.1

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

   6. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. frac-times N/A

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

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

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

      8. pow2 N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot {z}^{2}}{\color{blue}{t} \cdot \left(-t\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{{z}^{2} \cdot -1}{\color{blue}{t} \cdot \left(-t\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{{z}^{2}}{t} \cdot \color{blue}{\frac{-1}{-t}} \]

      12. pow2 N/A

          \[\leadsto \frac{z \cdot z}{t} \cdot \frac{-1}{-t} \]

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{z}{t} \cdot z\right) \cdot \frac{-1}{-t} \]

      15. lift-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 59.1

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

      20. lift-/.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{-1}{-t} \cdot z\right) \]

      21. metadata-eval N/A

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

      22. lift-neg.f64 N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(t\right)} \cdot z\right) \]

      23. frac-2neg N/A

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

      24. lower-/.f64 59.1

          \[\leadsto \frac{z}{t} \cdot \left(\frac{1}{t} \cdot z\right) \]

   8. Applied rewrites 59.1%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 56.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{z\_m \cdot z\_m}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \frac{z\_m}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= y 6e-222) (/ (/ (* z_m z_m) t) t) (* (/ z_m t) (/ z_m t))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (y <= 6e-222) {
		tmp = ((z_m * z_m) / t) / t;
	} else {
		tmp = (z_m / t) * (z_m / t);
	}
	return tmp;
}

Fortran

z_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, z_m, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 6d-222) then
        tmp = ((z_m * z_m) / t) / t
    else
        tmp = (z_m / t) * (z_m / t)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (y <= 6e-222) {
		tmp = ((z_m * z_m) / t) / t;
	} else {
		tmp = (z_m / t) * (z_m / t);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if y <= 6e-222:
		tmp = ((z_m * z_m) / t) / t
	else:
		tmp = (z_m / t) * (z_m / t)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (y <= 6e-222)
		tmp = Float64(Float64(Float64(z_m * z_m) / t) / t);
	else
		tmp = Float64(Float64(z_m / t) * Float64(z_m / t));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (y <= 6e-222)
		tmp = ((z_m * z_m) / t) / t;
	else
		tmp = (z_m / t) * (z_m / t);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[y, 6e-222], N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000059e-222

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l/ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. pow2 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 6.00000000000000059e-222 < y

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 59.1

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

   6. Applied rewrites 59.1%

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

Alternative 9: 56.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

2. Taylor expanded in x around 0

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

   1. pow2 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 52.1

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

4. Applied rewrites 52.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 59.1

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

6. Applied rewrites 59.1%

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

Alternative 10: 52.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

2. Taylor expanded in x around 0

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

   1. pow2 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 52.1

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

4. Applied rewrites 52.1%

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

Reproduce

herbie shell --seed 2025131 
(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))))