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

Percentage Accurate: 67.2% → 98.8%

Time: 13.0s

Alternatives: 6

Speedup: 1.0×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 10^{+292}:\\ \;\;\;\;z\_m \cdot \frac{\frac{z\_m}{t}}{t} + \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z\_m}{t} \cdot \left(\sqrt{z\_m} \cdot \frac{\sqrt{z\_m}}{t}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 1e+292)
   (+ (* z_m (/ (/ z_m t) t)) (/ (/ x y) (/ y x)))
   (+ (/ (* x (/ x y)) y) (* (/ z_m t) (* (sqrt z_m) (/ (sqrt z_m) t))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((z_m * z_m) / (t * t)) <= 1e+292) {
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	} else {
		tmp = ((x * (x / y)) / y) + ((z_m / t) * (sqrt(z_m) * (sqrt(z_m) / t)));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    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 (((z_m * z_m) / (t * t)) <= 1d+292) then
        tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x))
    else
        tmp = ((x * (x / y)) / y) + ((z_m / t) * (sqrt(z_m) * (sqrt(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 (((z_m * z_m) / (t * t)) <= 1e+292) {
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	} else {
		tmp = ((x * (x / y)) / y) + ((z_m / t) * (Math.sqrt(z_m) * (Math.sqrt(z_m) / t)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if ((z_m * z_m) / (t * t)) <= 1e+292:
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x))
	else:
		tmp = ((x * (x / y)) / y) + ((z_m / t) * (math.sqrt(z_m) * (math.sqrt(z_m) / t)))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t * t)) <= 1e+292)
		tmp = Float64(Float64(z_m * Float64(Float64(z_m / t) / t)) + Float64(Float64(x / y) / Float64(y / x)));
	else
		tmp = Float64(Float64(Float64(x * Float64(x / y)) / y) + Float64(Float64(z_m / t) * Float64(sqrt(z_m) * Float64(sqrt(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 (((z_m * z_m) / (t * t)) <= 1e+292)
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	else
		tmp = ((x * (x / y)) / y) + ((z_m / t) * (sqrt(z_m) * (sqrt(z_m) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 1e+292], N[(N[(z$95$m * N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(N[(z$95$m / t), $MachinePrecision] * N[(N[Sqrt[z$95$m], $MachinePrecision] * N[(N[Sqrt[z$95$m], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1e292

   1. Initial program 73.5%

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

      1. associate-/l* 79.1%

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

      2. fma-define 79.1%

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

      3. associate-/l* 81.7%

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

   3. Simplified 81.7%

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

      1. fma-undefine 81.7%

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

      2. associate-/l* 75.9%

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

      3. +-commutative 75.9%

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

      4. associate-/r* 77.9%

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

      5. associate-/l* 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate-*r/ 77.9%

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

      2. frac-times 99.6%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 1e292 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.3%

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

      1. add-sqr-sqrt 55.3%

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

      2. associate-*r/ 55.3%

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

      3. times-frac 69.5%

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

      4. sqrt-prod 36.8%

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

      5. add-sqr-sqrt 39.7%

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

      6. associate-*r/ 38.9%

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

      7. associate-*r/ 28.9%

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

      8. times-frac 49.9%

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

      9. sqrt-prod 43.5%

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

      10. add-sqr-sqrt 82.0%

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

   4. Applied egg-rr 82.0%

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

      1. add-sqr-sqrt 81.9%

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

      2. times-frac 81.9%

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

      3. sqrt-prod 48.8%

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

      4. add-sqr-sqrt 66.3%

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

      5. associate-*r/ 66.3%

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

      6. times-frac 79.8%

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

      7. sqrt-prod 56.2%

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

      8. add-sqr-sqrt 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-*r/ 92.4%

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

      3. add-sqr-sqrt 92.3%

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

      4. sqrt-prod 88.3%

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

      5. associate-*r* 85.8%

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

      6. *-commutative 85.8%

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

      7. sqrt-prod 45.2%

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

   8. Applied egg-rr 48.6%

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

      1. associate-*l* 50.2%

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

   10. Simplified 50.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+292}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t} + \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \left(\sqrt{z} \cdot \frac{\sqrt{z}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;z\_m \cdot \frac{\frac{z\_m}{t}}{t} + \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{1}{\frac{t}{z\_m} \cdot \frac{t}{z\_m}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 5e-22)
   (+ (* z_m (/ (/ z_m t) t)) (/ (/ x y) (/ y x)))
   (+ (/ (* x (/ x y)) y) (/ 1.0 (* (/ t z_m) (/ t z_m))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((z_m * z_m) / (t * t)) <= 5e-22) {
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	} else {
		tmp = ((x * (x / y)) / y) + (1.0 / ((t / z_m) * (t / z_m)));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    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 (((z_m * z_m) / (t * t)) <= 5d-22) then
        tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x))
    else
        tmp = ((x * (x / y)) / y) + (1.0d0 / ((t / z_m) * (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 (((z_m * z_m) / (t * t)) <= 5e-22) {
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	} else {
		tmp = ((x * (x / y)) / y) + (1.0 / ((t / z_m) * (t / z_m)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if ((z_m * z_m) / (t * t)) <= 5e-22:
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x))
	else:
		tmp = ((x * (x / y)) / y) + (1.0 / ((t / z_m) * (t / z_m)))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t * t)) <= 5e-22)
		tmp = Float64(Float64(z_m * Float64(Float64(z_m / t) / t)) + Float64(Float64(x / y) / Float64(y / x)));
	else
		tmp = Float64(Float64(Float64(x * Float64(x / y)) / y) + Float64(1.0 / Float64(Float64(t / z_m) * Float64(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 (((z_m * z_m) / (t * t)) <= 5e-22)
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	else
		tmp = ((x * (x / y)) / y) + (1.0 / ((t / z_m) * (t / z_m)));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e-22], N[(N[(z$95$m * N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(1.0 / N[(N[(t / z$95$m), $MachinePrecision] * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.99999999999999954e-22

   1. Initial program 74.6%

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

      1. associate-/l* 79.3%

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

      2. fma-define 79.3%

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

      3. associate-/l* 82.4%

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

   3. Simplified 82.4%

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

      1. fma-undefine 82.4%

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

      2. associate-/l* 77.5%

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

      3. +-commutative 77.5%

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

      4. associate-/r* 79.8%

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

      5. associate-/l* 84.9%

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

   6. Applied egg-rr 84.9%

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

      1. associate-*r/ 79.8%

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

      2. frac-times 99.6%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 4.99999999999999954e-22 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.1%

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

      1. add-sqr-sqrt 57.1%

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

      2. associate-*r/ 57.1%

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

      3. times-frac 69.1%

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

      4. sqrt-prod 34.1%

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

      5. add-sqr-sqrt 41.0%

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

      6. associate-*r/ 40.4%

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

      7. associate-*r/ 31.9%

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

      8. times-frac 49.6%

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

      9. sqrt-prod 39.8%

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

      10. add-sqr-sqrt 79.6%

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

   4. Applied egg-rr 79.6%

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

      1. add-sqr-sqrt 79.6%

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

      2. times-frac 79.6%

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

      3. sqrt-prod 46.4%

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

      4. add-sqr-sqrt 64.1%

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

      5. associate-*r/ 64.1%

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

      6. times-frac 78.7%

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

      7. sqrt-prod 54.9%

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

      8. add-sqr-sqrt 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. clear-num 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. times-frac 99.6%

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

      4. clear-num 99.6%

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

   8. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{1}{\frac{t}{z} \cdot \frac{t}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t} + \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{1}{\frac{t}{z} \cdot \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z\_m \cdot \frac{z\_m}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \frac{\frac{z\_m}{t}}{t} + \frac{\frac{x}{y}}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= t 2e-199)
   (+ (* (/ x y) (/ x y)) (/ (* z_m (/ z_m t)) t))
   (+ (* z_m (/ (/ z_m t) t)) (/ (/ x y) (/ y x)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (t <= 2e-199) {
		tmp = ((x / y) * (x / y)) + ((z_m * (z_m / t)) / t);
	} else {
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    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 (t <= 2d-199) then
        tmp = ((x / y) * (x / y)) + ((z_m * (z_m / t)) / t)
    else
        tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x))
    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 (t <= 2e-199) {
		tmp = ((x / y) * (x / y)) + ((z_m * (z_m / t)) / t);
	} else {
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if t <= 2e-199:
		tmp = ((x / y) * (x / y)) + ((z_m * (z_m / t)) / t)
	else:
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (t <= 2e-199)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + Float64(Float64(z_m * Float64(z_m / t)) / t));
	else
		tmp = Float64(Float64(z_m * Float64(Float64(z_m / t) / t)) + Float64(Float64(x / y) / Float64(y / x)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (t <= 2e-199)
		tmp = ((x / y) * (x / y)) + ((z_m * (z_m / t)) / t);
	else
		tmp = (z_m * ((z_m / t) / t)) + ((x / y) / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[t, 2e-199], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m * N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999996e-199

   1. Initial program 61.5%

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

      1. add-sqr-sqrt 61.5%

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

      2. associate-*r/ 61.5%

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

      3. times-frac 70.7%

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

      4. sqrt-prod 40.6%

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

      5. add-sqr-sqrt 52.9%

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

      6. associate-*r/ 52.3%

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

      7. associate-*r/ 46.5%

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

      8. times-frac 60.1%

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

      9. sqrt-prod 45.3%

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

      10. add-sqr-sqrt 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. times-frac 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 1.99999999999999996e-199 < t

   1. Initial program 71.9%

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

      1. associate-/l* 76.3%

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

      2. fma-define 76.3%

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

      3. associate-/l* 81.6%

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

   3. Simplified 81.6%

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

      1. fma-undefine 81.6%

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

      2. associate-/l* 77.2%

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

      3. +-commutative 77.2%

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

      4. associate-/r* 84.3%

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

      5. associate-/l* 88.7%

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

   6. Applied egg-rr 88.7%

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

      1. associate-*r/ 84.3%

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

      2. frac-times 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t} + \frac{\frac{x}{y}}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. associate-/l* 72.7%

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

   2. fma-define 72.7%

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

   3. associate-/l* 80.6%

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

3. Simplified 80.6%

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

   1. fma-undefine 80.6%

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

   2. associate-/l* 73.1%

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

   3. +-commutative 73.1%

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

   4. associate-/r* 78.6%

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

   5. associate-/l* 86.9%

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

6. Applied egg-rr 86.9%

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

   1. associate-*r/ 78.6%

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

   2. frac-times 96.4%

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

   3. clear-num 96.4%

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

   4. un-div-inv 96.5%

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

8. Applied egg-rr 96.5%

   \[\leadsto z \cdot \frac{\frac{z}{t}}{t} + \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]
9. Add Preprocessing

Alternative 5: 87.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. associate-/l* 72.7%

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

   2. fma-define 72.7%

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

   3. associate-/l* 80.6%

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

3. Simplified 80.6%

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

   1. fma-undefine 80.6%

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

   2. associate-/l* 73.1%

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

   3. +-commutative 73.1%

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

   4. associate-/r* 78.6%

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

   5. associate-/l* 86.9%

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

6. Applied egg-rr 86.9%

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

Alternative 6: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. associate-/l* 72.7%

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

   2. fma-define 72.7%

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

   3. associate-/l* 80.6%

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

3. Simplified 80.6%

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

   1. fma-undefine 80.6%

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

   2. associate-/l* 73.1%

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

   3. +-commutative 73.1%

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

   4. associate-/r* 78.6%

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

   5. associate-/l* 86.9%

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

6. Applied egg-rr 86.9%

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

   1. associate-/l/ 80.6%

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

8. Applied egg-rr 80.6%

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

9. Final simplification 80.6%

   \[\leadsto x \cdot \frac{x}{y \cdot y} + z \cdot \frac{z}{t \cdot t} \]
10. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	return pow((x / y), 2.0) + pow((z / t), 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) ** 2.0d0) + ((z / t) ** 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((x / y), 2.0) + Math.pow((z / t), 2.0);
}

Python

def code(x, y, z, t):
	return math.pow((x / y), 2.0) + math.pow((z / t), 2.0)

Julia

function code(x, y, z, t)
	return Float64((Float64(x / y) ^ 2.0) + (Float64(z / t) ^ 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) ^ 2.0) + ((z / t) ^ 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024097 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :alt
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

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