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

Percentage Accurate: 66.0% → 97.0%

Time: 10.7s

Alternatives: 15

Speedup: 0.8×

• 47.7% 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: 66.0% 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: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+248}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + \frac{1}{t} \cdot \left(z\_m \cdot \frac{z\_m}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(\frac{1}{t} \cdot \frac{z\_m}{t}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.5e+248)
   (+ (/ (/ x y) (/ y x)) (* (/ 1.0 t) (* z_m (/ z_m t))))
   (* z_m (* (/ 1.0 t) (/ z_m t)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.5e+248) {
		tmp = ((x / y) / (y / x)) + ((1.0 / t) * (z_m * (z_m / t)));
	} else {
		tmp = z_m * ((1.0 / t) * (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 <= 3.5d+248) then
        tmp = ((x / y) / (y / x)) + ((1.0d0 / t) * (z_m * (z_m / t)))
    else
        tmp = z_m * ((1.0d0 / 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 (z_m <= 3.5e+248) {
		tmp = ((x / y) / (y / x)) + ((1.0 / t) * (z_m * (z_m / t)));
	} else {
		tmp = z_m * ((1.0 / t) * (z_m / t));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if z_m <= 3.5e+248:
		tmp = ((x / y) / (y / x)) + ((1.0 / t) * (z_m * (z_m / t)))
	else:
		tmp = z_m * ((1.0 / t) * (z_m / t))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.5e+248)
		tmp = Float64(Float64(Float64(x / y) / Float64(y / x)) + Float64(Float64(1.0 / t) * Float64(z_m * Float64(z_m / t))));
	else
		tmp = Float64(z_m * Float64(Float64(1.0 / 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 (z_m <= 3.5e+248)
		tmp = ((x / y) / (y / x)) + ((1.0 / t) * (z_m * (z_m / t)));
	else
		tmp = z_m * ((1.0 / 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[z$95$m, 3.5e+248], N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / t), $MachinePrecision] * N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(N[(1.0 / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.50000000000000022e248

   1. Initial program 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.4

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

   6. Applied rewrites 97.4%

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

   if 3.50000000000000022e248 < z

   1. Initial program 42.9%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+248}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + \frac{1}{t} \cdot \left(z \cdot \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{1}{t} \cdot \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-138}:\\ \;\;\;\;t\_1 + \frac{\frac{z\_m}{t}}{\frac{t}{z\_m}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z\_m \cdot \frac{z\_m}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + \frac{z\_m \cdot z\_m}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 5e-138)
     (+ t_1 (/ (/ z_m t) (/ t z_m)))
     (if (<= t_1 INFINITY)
       (+ (* x (/ x (* y y))) (/ (* z_m (/ z_m t)) t))
       (+ (/ (/ x y) (/ y x)) (/ (* z_m z_m) (* t t)))))))

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 <= 5e-138) {
		tmp = t_1 + ((z_m / t) / (t / z_m));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * (x / (y * y))) + ((z_m * (z_m / t)) / t);
	} else {
		tmp = ((x / y) / (y / x)) + ((z_m * z_m) / (t * t));
	}
	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 <= 5e-138) {
		tmp = t_1 + ((z_m / t) / (t / z_m));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * (x / (y * y))) + ((z_m * (z_m / t)) / t);
	} else {
		tmp = ((x / y) / (y / x)) + ((z_m * z_m) / (t * t));
	}
	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 <= 5e-138:
		tmp = t_1 + ((z_m / t) / (t / z_m))
	elif t_1 <= math.inf:
		tmp = (x * (x / (y * y))) + ((z_m * (z_m / t)) / t)
	else:
		tmp = ((x / y) / (y / x)) + ((z_m * z_m) / (t * t))
	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 <= 5e-138)
		tmp = Float64(t_1 + Float64(Float64(z_m / t) / Float64(t / z_m)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * Float64(x / Float64(y * y))) + Float64(Float64(z_m * Float64(z_m / t)) / t));
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(y / x)) + Float64(Float64(z_m * z_m) / Float64(t * t)));
	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 <= 5e-138)
		tmp = t_1 + ((z_m / t) / (t / z_m));
	elseif (t_1 <= Inf)
		tmp = (x * (x / (y * y))) + ((z_m * (z_m / t)) / t);
	else
		tmp = ((x / y) / (y / x)) + ((z_m * z_m) / (t * t));
	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, 5e-138], N[(t$95$1 + N[(N[(z$95$m / t), $MachinePrecision] / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 4.99999999999999989e-138

   1. Initial program 64.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 4.99999999999999989e-138 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 87.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.1

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

   4. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.5

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

   4. Applied rewrites 79.5%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + \frac{z \cdot z}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-138}:\\ \;\;\;\;t\_1 + \frac{\frac{z\_m}{t}}{\frac{t}{z\_m}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z\_m \cdot \frac{z\_m}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z\_m \cdot z\_m}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 5e-138)
     (+ t_1 (/ (/ z_m t) (/ t z_m)))
     (if (<= t_1 INFINITY)
       (+ (* x (/ x (* y y))) (/ (* z_m (/ z_m t)) t))
       (fma (/ x y) (/ x y) (/ (* z_m z_m) (* t t)))))))

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 <= 5e-138) {
		tmp = t_1 + ((z_m / t) / (t / z_m));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * (x / (y * y))) + ((z_m * (z_m / t)) / t);
	} else {
		tmp = fma((x / y), (x / y), ((z_m * z_m) / (t * t)));
	}
	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 <= 5e-138)
		tmp = Float64(t_1 + Float64(Float64(z_m / t) / Float64(t / z_m)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * Float64(x / Float64(y * y))) + Float64(Float64(z_m * Float64(z_m / t)) / t));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z_m * z_m) / Float64(t * t)));
	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, 5e-138], N[(t$95$1 + N[(N[(z$95$m / t), $MachinePrecision] / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 4.99999999999999989e-138

   1. Initial program 64.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 4.99999999999999989e-138 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 87.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 94.1

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

   4. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.4

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

   4. Applied rewrites 79.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t}, \frac{z\_m}{t}, t\_1\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z\_m \cdot \frac{z\_m}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z\_m \cdot z\_m}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 5e+48)
     (fma (/ z_m t) (/ z_m t) t_1)
     (if (<= t_1 INFINITY)
       (+ (* x (/ x (* y y))) (/ (* z_m (/ z_m t)) t))
       (fma (/ x y) (/ x y) (/ (* z_m z_m) (* t t)))))))

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 <= 5e+48) {
		tmp = fma((z_m / t), (z_m / t), t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * (x / (y * y))) + ((z_m * (z_m / t)) / t);
	} else {
		tmp = fma((x / y), (x / y), ((z_m * z_m) / (t * t)));
	}
	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 <= 5e+48)
		tmp = fma(Float64(z_m / t), Float64(z_m / t), t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * Float64(x / Float64(y * y))) + Float64(Float64(z_m * Float64(z_m / t)) / t));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z_m * z_m) / Float64(t * t)));
	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, 5e+48], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z$95$m * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 4.99999999999999973e48

   1. Initial program 70.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

   if 4.99999999999999973e48 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 97.3

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

   6. Applied rewrites 97.3%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.4

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

   4. Applied rewrites 79.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{-202}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \frac{z\_m}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t t))))
   (if (<= t_1 1e-202)
     (* (/ x y) (* x (/ 1.0 y)))
     (if (<= t_1 2e+190) (fma (/ x (* y y)) x t_1) (* (/ z_m t) (/ z_m t))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double tmp;
	if (t_1 <= 1e-202) {
		tmp = (x / y) * (x * (1.0 / y));
	} else if (t_1 <= 2e+190) {
		tmp = fma((x / (y * y)), x, t_1);
	} else {
		tmp = (z_m / t) * (z_m / t);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e-202)
		tmp = Float64(Float64(x / y) * Float64(x * Float64(1.0 / y)));
	elseif (t_1 <= 2e+190)
		tmp = fma(Float64(x / Float64(y * y)), x, t_1);
	else
		tmp = Float64(Float64(z_m / t) * Float64(z_m / t));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-202], N[(N[(x / y), $MachinePrecision] * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+190], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1e-202

   1. Initial program 76.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 94.2%

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

   9. Applied rewrites 94.3%

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

   if 1e-202 < (/.f64 (*.f64 z z) (*.f64 t t)) < 2.0000000000000001e190

   1. Initial program 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if 2.0000000000000001e190 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 52.8%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 86.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{-202}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t \cdot t}\\ t_2 := x \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z\_m \cdot \frac{z\_m}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t t))) (t_2 (* x (/ (/ x y) y))))
   (if (<= t_1 5e-161)
     t_2
     (if (<= t_1 INFINITY) (* z_m (/ z_m (* t t))) t_2))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double t_2 = x * ((x / y) / y);
	double tmp;
	if (t_1 <= 5e-161) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z_m * (z_m / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double t_2 = x * ((x / y) / y);
	double tmp;
	if (t_1 <= 5e-161) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z_m * (z_m / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) / (t * t)
	t_2 = x * ((x / y) / y)
	tmp = 0
	if t_1 <= 5e-161:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = z_m * (z_m / (t * t))
	else:
		tmp = t_2
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t * t))
	t_2 = Float64(x * Float64(Float64(x / y) / y))
	tmp = 0.0
	if (t_1 <= 5e-161)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(z_m * Float64(z_m / Float64(t * t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) / (t * t);
	t_2 = x * ((x / y) / y);
	tmp = 0.0;
	if (t_1 <= 5e-161)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = z_m * (z_m / (t * t));
	else
		tmp = t_2;
	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[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-161], t$95$2, If[LessEqual[t$95$1, Infinity], N[(z$95$m * N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e-161 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 75.8%

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

   if 4.9999999999999999e-161 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

Alternative 7: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{-183}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z\_m \cdot \frac{z\_m}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t t))))
   (if (<= t_1 1e-183)
     (* x (/ x (* y y)))
     (if (<= t_1 INFINITY)
       (* z_m (/ z_m (* t t)))
       (* x (* x (/ 1.0 (* y y))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double tmp;
	if (t_1 <= 1e-183) {
		tmp = x * (x / (y * y));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z_m * (z_m / (t * t));
	} else {
		tmp = x * (x * (1.0 / (y * y)));
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double tmp;
	if (t_1 <= 1e-183) {
		tmp = x * (x / (y * y));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z_m * (z_m / (t * t));
	} else {
		tmp = x * (x * (1.0 / (y * y)));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) / (t * t)
	tmp = 0
	if t_1 <= 1e-183:
		tmp = x * (x / (y * y))
	elif t_1 <= math.inf:
		tmp = z_m * (z_m / (t * t))
	else:
		tmp = x * (x * (1.0 / (y * y)))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e-183)
		tmp = Float64(x * Float64(x / Float64(y * y)));
	elseif (t_1 <= Inf)
		tmp = Float64(z_m * Float64(z_m / Float64(t * t)));
	else
		tmp = Float64(x * Float64(x * Float64(1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) / (t * t);
	tmp = 0.0;
	if (t_1 <= 1e-183)
		tmp = x * (x / (y * y));
	elseif (t_1 <= Inf)
		tmp = z_m * (z_m / (t * t));
	else
		tmp = x * (x * (1.0 / (y * y)));
	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[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-183], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(z$95$m * N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.00000000000000001e-183

   1. Initial program 77.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if 1.00000000000000001e-183 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 36.3

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

   5. Applied rewrites 36.3%

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

   7. Applied rewrites 36.3%

      \[\leadsto x \cdot \left(\frac{1}{y \cdot y} \cdot \color{blue}{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{-183}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 0.6× speedup

Math

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

FPCore

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

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 <= ((double) INFINITY)) {
		tmp = fma((z_m / t), (z_m / t), t_1);
	} else {
		tmp = fma((x / y), (x / y), ((z_m * z_m) / (t * t)));
	}
	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 <= Inf)
		tmp = fma(Float64(z_m / t), Float64(z_m / t), t_1);
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z_m * z_m) / Float64(t * t)));
	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, Infinity], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.4

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

   4. Applied rewrites 79.4%

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

Alternative 9: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \frac{z\_m}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t t))))
   (if (<= t_1 5e+281) (fma (/ x y) (/ x y) t_1) (* (/ z_m t) (/ z_m t)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double tmp;
	if (t_1 <= 5e+281) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = (z_m / t) * (z_m / t);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+281)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = Float64(Float64(z_m / t) * Float64(z_m / t));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+281], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000016e281

   1. Initial program 78.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   if 5.00000000000000016e281 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 50.4%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 85.5%

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

Alternative 10: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t \cdot t}\\ t_2 := x \cdot \frac{x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{-183}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z\_m \cdot \frac{z\_m}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t t))) (t_2 (* x (/ x (* y y)))))
   (if (<= t_1 1e-183)
     t_2
     (if (<= t_1 INFINITY) (* z_m (/ z_m (* t t))) t_2))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double t_2 = x * (x / (y * y));
	double tmp;
	if (t_1 <= 1e-183) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z_m * (z_m / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double t_2 = x * (x / (y * y));
	double tmp;
	if (t_1 <= 1e-183) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z_m * (z_m / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (z_m * z_m) / (t * t)
	t_2 = x * (x / (y * y))
	tmp = 0
	if t_1 <= 1e-183:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = z_m * (z_m / (t * t))
	else:
		tmp = t_2
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t * t))
	t_2 = Float64(x * Float64(x / Float64(y * y)))
	tmp = 0.0
	if (t_1 <= 1e-183)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(z_m * Float64(z_m / Float64(t * t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (z_m * z_m) / (t * t);
	t_2 = x * (x / (y * y));
	tmp = 0.0;
	if (t_1 <= 1e-183)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = z_m * (z_m / (t * t));
	else
		tmp = t_2;
	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[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-183], t$95$2, If[LessEqual[t$95$1, Infinity], N[(z$95$m * N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.00000000000000001e-183 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if 1.00000000000000001e-183 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 11: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \frac{z\_m}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 5e-161)
   (* (/ x y) (* x (/ 1.0 y)))
   (* (/ 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 (((z_m * z_m) / (t * t)) <= 5e-161) {
		tmp = (x / y) * (x * (1.0 / y));
	} else {
		tmp = (z_m / t) * (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)) <= 5d-161) then
        tmp = (x / y) * (x * (1.0d0 / y))
    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 (((z_m * z_m) / (t * t)) <= 5e-161) {
		tmp = (x / y) * (x * (1.0 / y));
	} 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 ((z_m * z_m) / (t * t)) <= 5e-161:
		tmp = (x / y) * (x * (1.0 / y))
	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 (Float64(Float64(z_m * z_m) / Float64(t * t)) <= 5e-161)
		tmp = Float64(Float64(x / y) * Float64(x * Float64(1.0 / y)));
	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 (((z_m * z_m) / (t * t)) <= 5e-161)
		tmp = (x / y) * (x * (1.0 / y));
	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[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e-161], N[(N[(x / y), $MachinePrecision] * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e-161

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 93.5%

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

   9. Applied rewrites 93.5%

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

   if 4.9999999999999999e-161 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 83.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \frac{z\_m}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 5e-161)
   (* (/ x y) (/ x y))
   (* (/ 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 (((z_m * z_m) / (t * t)) <= 5e-161) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z_m / t) * (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)) <= 5d-161) then
        tmp = (x / y) * (x / y)
    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 (((z_m * z_m) / (t * t)) <= 5e-161) {
		tmp = (x / y) * (x / y);
	} 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 ((z_m * z_m) / (t * t)) <= 5e-161:
		tmp = (x / y) * (x / y)
	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 (Float64(Float64(z_m * z_m) / Float64(t * t)) <= 5e-161)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	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 (((z_m * z_m) / (t * t)) <= 5e-161)
		tmp = (x / y) * (x / y);
	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[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e-161], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e-161

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 93.5%

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

   if 4.9999999999999999e-161 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 83.8%

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

Alternative 13: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \frac{\frac{z\_m}{t}}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 5e-161)
   (* (/ x y) (/ x y))
   (* z_m (/ (/ z_m t) 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)) <= 5e-161) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = z_m * ((z_m / t) / 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)) <= 5d-161) then
        tmp = (x / y) * (x / y)
    else
        tmp = z_m * ((z_m / t) / 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)) <= 5e-161) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = z_m * ((z_m / t) / 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)) <= 5e-161:
		tmp = (x / y) * (x / y)
	else:
		tmp = z_m * ((z_m / t) / 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)) <= 5e-161)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(z_m * Float64(Float64(z_m / t) / 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)) <= 5e-161)
		tmp = (x / y) * (x / y);
	else
		tmp = z_m * ((z_m / t) / 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], 5e-161], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e-161

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 93.5%

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

   if 4.9999999999999999e-161 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 81.0%

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

Alternative 14: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \frac{\frac{z\_m}{t}}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 5e-161)
   (* 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) {
	double tmp;
	if (((z_m * z_m) / (t * t)) <= 5e-161) {
		tmp = x * ((x / y) / y);
	} else {
		tmp = z_m * ((z_m / t) / 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)) <= 5d-161) then
        tmp = x * ((x / y) / y)
    else
        tmp = z_m * ((z_m / t) / 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)) <= 5e-161) {
		tmp = x * ((x / y) / y);
	} else {
		tmp = z_m * ((z_m / t) / 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)) <= 5e-161:
		tmp = x * ((x / y) / y)
	else:
		tmp = z_m * ((z_m / t) / 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)) <= 5e-161)
		tmp = Float64(x * Float64(Float64(x / y) / y));
	else
		tmp = Float64(z_m * Float64(Float64(z_m / t) / 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)) <= 5e-161)
		tmp = x * ((x / y) / y);
	else
		tmp = z_m * ((z_m / t) / 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], 5e-161], N[(x * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e-161

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 84.9%

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

   if 4.9999999999999999e-161 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 81.0%

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

Alternative 15: 51.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 52.6

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

5. Applied rewrites 52.6%

   \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× 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 2024220 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

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

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