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

Percentage Accurate: 66.7% → 99.7%

Time: 12.5s

Alternatives: 16

Speedup: 0.8×

• 48.2% 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: 66.7% 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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   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 82.1

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

4. Applied rewrites 82.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. lift-/.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 2: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-305)
     (* (/ x y) (/ x y))
     (if (<= t_1 1e+285) (fma (/ x (* y y)) x t_1) (/ (/ z t) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= 1e+285) {
		tmp = fma((x / (y * y)), x, t_1);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-305)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= 1e+285)
		tmp = fma(Float64(x / Float64(y * y)), x, t_1);
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-305], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+285], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999999e-305

   1. Initial program 70.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 92.8%

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

   if 1.99999999999999999e-305 < (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999998e284

   1. Initial program 80.1%

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

      1. lift-+.f64 N/A

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

      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 84.6

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

   4. Applied rewrites 84.6%

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

   if 9.9999999999999998e284 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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 73.4

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

   7. Applied rewrites 73.4%

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

   9. Applied rewrites 84.6%

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

Alternative 3: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-305)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY) (* z (/ z (* t t))) (/ x (* y (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x / (y * (y / x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-305:
		tmp = (x / y) * (x / y)
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	else:
		tmp = x / (y * (y / x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-305)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	else
		tmp = Float64(x / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 2e-305)
		tmp = (x / y) * (x / y);
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	else
		tmp = x / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-305], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999999e-305

   1. Initial program 70.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 92.8%

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

   if 1.99999999999999999e-305 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 80.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   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 35.9

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

   5. Applied rewrites 35.9%

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

   7. Applied rewrites 35.9%

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

   9. Applied rewrites 42.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-305}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-305)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY) (* z (/ z (* t t))) (* x (/ (/ x y) y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x * ((x / y) / y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x * ((x / y) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-305:
		tmp = (x / y) * (x / y)
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	else:
		tmp = x * ((x / y) / y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-305)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	else
		tmp = Float64(x * Float64(Float64(x / y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 2e-305)
		tmp = (x / y) * (x / y);
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	else
		tmp = x * ((x / y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-305], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999999e-305

   1. Initial program 70.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 92.8%

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

   if 1.99999999999999999e-305 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 80.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   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 35.9

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

   5. Applied rewrites 35.9%

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

   7. Applied rewrites 42.1%

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

Alternative 5: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := x \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* x (/ (/ x y) y))))
   (if (<= t_1 2e-305) t_2 (if (<= t_1 INFINITY) (* z (/ z (* t t))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = x * ((x / y) / y);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = x * ((x / y) / y);
	double tmp;
	if (t_1 <= 2e-305) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = x * ((x / y) / y)
	tmp = 0
	if t_1 <= 2e-305:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(x * Float64(Float64(x / y) / y))
	tmp = 0.0
	if (t_1 <= 2e-305)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = x * ((x / y) / y);
	tmp = 0.0;
	if (t_1 <= 2e-305)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $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, 2e-305], t$95$2, If[LessEqual[t$95$1, Infinity], N[(z * N[(z / 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)) < 1.99999999999999999e-305 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 51.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 75.5%

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

   if 1.99999999999999999e-305 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 80.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 6: 94.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(z \cdot z\right) \cdot \frac{1}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 40000000000.0)
   (fma (/ x y) (/ x y) (* (* z z) (/ 1.0 (* t t))))
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 40000000000.0) {
		tmp = fma((x / y), (x / y), ((z * z) * (1.0 / (t * t))));
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 40000000000.0)
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z * z) * Float64(1.0 / Float64(t * t))));
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 40000000000.0], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] * N[(1.0 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4e10

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 94.6

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

   6. Applied rewrites 94.6%

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

   if 4e10 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 62.1%

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

      1. lift-+.f64 N/A

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

      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 70.4

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

   4. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(z \cdot z\right) \cdot \frac{1}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{1}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e+14)
     (* z (/ z (* t t)))
     (if (<= t_1 INFINITY)
       (* x (* x (/ 1.0 (* y y))))
       (* z (* z (/ 1.0 (* t t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e+14) {
		tmp = z * (z / (t * t));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x * (1.0 / (y * y)));
	} else {
		tmp = z * (z * (1.0 / (t * t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e+14) {
		tmp = z * (z / (t * t));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x * (1.0 / (y * y)));
	} else {
		tmp = z * (z * (1.0 / (t * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if t_1 <= 2e+14:
		tmp = z * (z / (t * t))
	elif t_1 <= math.inf:
		tmp = x * (x * (1.0 / (y * y)))
	else:
		tmp = z * (z * (1.0 / (t * t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e+14)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x * Float64(1.0 / Float64(y * y))));
	else
		tmp = Float64(z * Float64(z * Float64(1.0 / Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	tmp = 0.0;
	if (t_1 <= 2e+14)
		tmp = z * (z / (t * t));
	elseif (t_1 <= Inf)
		tmp = x * (x * (1.0 / (y * y)));
	else
		tmp = z * (z * (1.0 / (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+14], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x * N[(x * N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(1.0 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2e14

   1. Initial program 72.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   if 2e14 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 79.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 89.1

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

   5. Applied rewrites 89.1%

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

   7. Applied rewrites 89.1%

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

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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 45.4

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 45.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{1}{t \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := z \cdot \frac{z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* z (/ z (* t t)))))
   (if (<= t_1 2e+14)
     t_2
     (if (<= t_1 INFINITY) (* x (* x (/ 1.0 (* y y)))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * (z / (t * t));
	double tmp;
	if (t_1 <= 2e+14) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x * (1.0 / (y * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * (z / (t * t));
	double tmp;
	if (t_1 <= 2e+14) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x * (1.0 / (y * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = z * (z / (t * t))
	tmp = 0
	if t_1 <= 2e+14:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = x * (x * (1.0 / (y * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(z * Float64(z / Float64(t * t)))
	tmp = 0.0
	if (t_1 <= 2e+14)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x * Float64(1.0 / Float64(y * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = z * (z / (t * t));
	tmp = 0.0;
	if (t_1 <= 2e+14)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = x * (x * (1.0 / (y * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+14], t$95$2, If[LessEqual[t$95$1, Infinity], N[(x * N[(x * N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2e14 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 58.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 68.2

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

   5. Applied rewrites 68.2%

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

   if 2e14 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 79.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 89.1

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

   5. Applied rewrites 89.1%

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

   7. Applied rewrites 89.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 40000000000.0)
     (fma (/ x y) (/ x y) t_1)
     (fma (/ x (* y y)) x (* (/ z t) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 40000000000.0) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 40000000000.0)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 40000000000.0], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4e10

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 94.6

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

   4. Applied rewrites 94.6%

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

   if 4e10 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 62.1%

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

      1. lift-+.f64 N/A

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

      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 70.4

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

   4. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

Alternative 10: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+285) (fma (/ x y) (/ x y) t_1) (/ (/ z t) (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+285) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+285)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+285], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999998e284

   1. Initial program 74.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 94.9

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

   4. Applied rewrites 94.9%

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

   if 9.9999999999999998e284 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.4

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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 73.4

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

   7. Applied rewrites 73.4%

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

   9. Applied rewrites 84.6%

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

Alternative 11: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := z \cdot \frac{z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* z (/ z (* t t)))))
   (if (<= t_1 2e+14) t_2 (if (<= t_1 INFINITY) (* x (/ x (* y y))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * (z / (t * t));
	double tmp;
	if (t_1 <= 2e+14) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x / (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * (z / (t * t));
	double tmp;
	if (t_1 <= 2e+14) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x / (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = z * (z / (t * t))
	tmp = 0
	if t_1 <= 2e+14:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = x * (x / (y * y))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(z * Float64(z / Float64(t * t)))
	tmp = 0.0
	if (t_1 <= 2e+14)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x / Float64(y * y)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = z * (z / (t * t));
	tmp = 0.0;
	if (t_1 <= 2e+14)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = x * (x / (y * y));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+14], t$95$2, If[LessEqual[t$95$1, Infinity], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2e14 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 58.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 68.2

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

   5. Applied rewrites 68.2%

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

   if 2e14 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 79.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 89.1

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

   5. Applied rewrites 89.1%

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

Alternative 12: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e+16) (/ (/ z t) (/ t z)) (* (/ x y) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+16) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 1d+16) then
        tmp = (z / t) / (t / z)
    else
        tmp = (x / y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+16) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 1e+16:
		tmp = (z / t) / (t / z)
	else:
		tmp = (x / y) * (x / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e+16)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 1e+16)
		tmp = (z / t) / (t / z);
	else
		tmp = (x / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e+16], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1e16

   1. Initial program 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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 73.2

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 87.9%

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

   if 1e16 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 61.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 80.3%

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

Alternative 13: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   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 82.1

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

4. Applied rewrites 82.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 14: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{+16}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e+16) (* (/ z t) (/ z t)) (* (/ x y) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+16) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 1d+16) then
        tmp = (z / t) * (z / t)
    else
        tmp = (x / y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+16) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 1e+16:
		tmp = (z / t) * (z / t)
	else:
		tmp = (x / y) * (x / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e+16)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 1e+16)
		tmp = (z / t) * (z / t);
	else
		tmp = (x / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e+16], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1e16

   1. Initial program 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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 73.2

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 87.7%

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

   if 1e16 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 61.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 80.3%

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

Alternative 15: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{+16}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1e+16) (* z (/ (/ z t) t)) (* (/ x y) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+16) {
		tmp = z * ((z / t) / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 1d+16) then
        tmp = z * ((z / t) / t)
    else
        tmp = (x / y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1e+16) {
		tmp = z * ((z / t) / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 1e+16:
		tmp = z * ((z / t) / t)
	else:
		tmp = (x / y) * (x / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1e+16)
		tmp = Float64(z * Float64(Float64(z / t) / t));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 1e+16)
		tmp = z * ((z / t) / t);
	else
		tmp = (x / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1e+16], N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1e16

   1. Initial program 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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 73.2

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 83.2%

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

   if 1e16 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 61.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 80.3%

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

Alternative 16: 52.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 52.0

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

5. Applied rewrites 52.0%

   \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
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 2024223 
(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))))