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

Percentage Accurate: 67.4% → 96.0%

Time: 10.0s

Alternatives: 11

Speedup: 0.8×

• 48.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% 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: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{\frac{y \cdot y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e+121)
   (fma (/ x y) (/ x y) (* (/ z (* t t)) z))
   (fma (/ z t) (/ z t) (/ x (/ (* y y) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e+121) {
		tmp = fma((x / y), (x / y), ((z / (t * t)) * z));
	} else {
		tmp = fma((z / t), (z / t), (x / ((y * y) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e+121)
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z / Float64(t * t)) * z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(x / Float64(Float64(y * y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e+121], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x / N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.00000000000000007e121

   1. Initial program 71.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 94.7

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if 2.00000000000000007e121 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. +-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 85.2

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.7

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

   6. Applied rewrites 97.7%

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

Alternative 2: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x} \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-277)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY)
       (fma (/ z (* t t)) z (* (/ x (* y y)) x))
       (/ x (* (/ y x) 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-277) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((z / (t * t)), z, ((x / (y * y)) * x));
	} else {
		tmp = x / ((y / x) * 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-277)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / Float64(y * y)) * x));
	else
		tmp = Float64(x / Float64(Float64(y / x) * y));
	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-277], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999994e-277

   1. Initial program 69.9%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 93.2%

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

   if 1.99999999999999994e-277 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 75.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. +-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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 78.1

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 89.7

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

   4. Applied rewrites 89.7%

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 47.2%

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

   9. Applied rewrites 55.9%

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

Alternative 3: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x} \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-118)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY) (/ (* (/ z t) z) t) (/ x (* (/ y x) 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-118) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((z / t) * z) / t;
	} else {
		tmp = x / ((y / x) * 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-118) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((z / t) * z) / t;
	} else {
		tmp = x / ((y / x) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-118:
		tmp = (x / y) * (x / y)
	elif t_1 <= math.inf:
		tmp = ((z / t) * z) / t
	else:
		tmp = x / ((y / x) * 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-118)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(z / t) * z) / t);
	else
		tmp = Float64(x / Float64(Float64(y / x) * 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-118)
		tmp = (x / y) * (x / y);
	elseif (t_1 <= Inf)
		tmp = ((z / t) * z) / t;
	else
		tmp = x / ((y / x) * 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-118], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999997e-118

   1. Initial program 70.3%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 90.0%

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

   if 1.99999999999999997e-118 < (/.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. 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 82.9

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

   4. Applied rewrites 82.9%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.0

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

   9. Applied rewrites 81.0%

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

   11. Applied rewrites 87.8%

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

   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 t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 47.2%

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

   9. Applied rewrites 55.9%

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

Alternative 4: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x} \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-118)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY) t_1 (/ x (* (/ y x) 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-118) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x / ((y / x) * 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-118) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = x / ((y / x) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-118:
		tmp = (x / y) * (x / y)
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = x / ((y / x) * 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-118)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(Float64(y / x) * 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-118)
		tmp = (x / y) * (x / y);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = x / ((y / x) * 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-118], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(x / N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999997e-118

   1. Initial program 70.3%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 90.0%

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

   if 1.99999999999999997e-118 < (/.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. 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 82.9

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

   4. Applied rewrites 82.9%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.0

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

   9. Applied rewrites 81.0%

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

   11. Applied rewrites 82.5%

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

   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 t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 47.2%

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

   9. Applied rewrites 55.9%

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

Alternative 5: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* (/ x y) (/ x y))))
   (if (<= t_1 2e-118) t_2 (if (<= t_1 INFINITY) t_1 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = (x / y) * (x / y);
	double tmp;
	if (t_1 <= 2e-118) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} 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 / y) * (x / y);
	double tmp;
	if (t_1 <= 2e-118) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = (x / y) * (x / y)
	tmp = 0
	if t_1 <= 2e-118:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = t_1
	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(Float64(x / y) * Float64(x / y))
	tmp = 0.0
	if (t_1 <= 2e-118)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_1;
	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 / y) * (x / y);
	tmp = 0.0;
	if (t_1 <= 2e-118)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_1;
	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[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-118], t$95$2, If[LessEqual[t$95$1, Infinity], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999997e-118 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 52.3%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 81.2%

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

   if 1.99999999999999997e-118 < (/.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. 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 82.9

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

   4. Applied rewrites 82.9%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.0

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

   9. Applied rewrites 81.0%

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

   11. Applied rewrites 82.5%

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

Alternative 6: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \left(\frac{1}{t \cdot t} \cdot z\right) \cdot z\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (* (/ 1.0 (* t t)) z) z)))
   (if (<= t_1 2e-117) t_2 (if (<= t_1 INFINITY) (* (/ x (* y y)) x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = ((1.0 / (t * t)) * z) * z;
	double tmp;
	if (t_1 <= 2e-117) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / (y * y)) * x;
	} 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 = ((1.0 / (t * t)) * z) * z;
	double tmp;
	if (t_1 <= 2e-117) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = ((1.0 / (t * t)) * z) * z
	tmp = 0
	if t_1 <= 2e-117:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / (y * y)) * x
	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(Float64(Float64(1.0 / Float64(t * t)) * z) * z)
	tmp = 0.0
	if (t_1 <= 2e-117)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / Float64(y * y)) * x);
	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 = ((1.0 / (t * t)) * z) * z;
	tmp = 0.0;
	if (t_1 <= 2e-117)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / (y * y)) * x;
	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[(N[(N[(1.0 / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-117], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2.00000000000000006e-117 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 51.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 68.4

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

   4. Applied rewrites 68.4%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 64.9

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

   9. Applied rewrites 64.9%

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

   11. Applied rewrites 65.0%

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

   if 2.00000000000000006e-117 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.2%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 7: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e+121)
   (fma (/ x y) (/ x y) (* (/ z (* t t)) z))
   (fma (/ z t) (/ z t) (* (/ x (* y y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e+121) {
		tmp = fma((x / y), (x / y), ((z / (t * t)) * z));
	} else {
		tmp = fma((z / t), (z / t), ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e+121)
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z / Float64(t * t)) * z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e+121], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.00000000000000007e121

   1. Initial program 71.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 94.7

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if 2.00000000000000007e121 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. +-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 85.2

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

Alternative 8: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 0:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t \cdot t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 0.0)
   (/ (/ z t) (/ t z))
   (fma (/ x y) (/ x y) (* (/ z (* t t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 0.0) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = fma((x / y), (x / y), ((z / (t * t)) * z));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 0.0

   1. Initial program 67.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. 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 90.8

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

   4. Applied rewrites 90.8%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.0

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

   9. Applied rewrites 73.0%

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

   11. Applied rewrites 92.9%

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

   if 0.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 61.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. 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 83.4

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 91.3

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

   4. Applied rewrites 91.3%

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

Alternative 9: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t \cdot t} \cdot z\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z (* t t)) z)))
   (if (<= t_1 2e-117) t_2 (if (<= t_1 INFINITY) (* (/ x (* y y)) x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / (t * t)) * z;
	double tmp;
	if (t_1 <= 2e-117) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / (y * y)) * x;
	} 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 / (t * t)) * z;
	double tmp;
	if (t_1 <= 2e-117) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / (t * t)) * z
	tmp = 0
	if t_1 <= 2e-117:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / (y * y)) * x
	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(Float64(z / Float64(t * t)) * z)
	tmp = 0.0
	if (t_1 <= 2e-117)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / Float64(y * y)) * x);
	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 / (t * t)) * z;
	tmp = 0.0;
	if (t_1 <= 2e-117)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / (y * y)) * x;
	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[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-117], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2.00000000000000006e-117 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 51.6%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if 2.00000000000000006e-117 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.2%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 10: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 4 \cdot 10^{-115}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 4e-115) (* (/ 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)) <= 4e-115) {
		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)) <= 4d-115) 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)) <= 4e-115) {
		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)) <= 4e-115:
		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)) <= 4e-115)
		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)) <= 4e-115)
		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], 4e-115], 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)) < 4.0000000000000002e-115

   1. Initial program 68.4%

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

      1. lift-/.f64 N/A

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

      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 90.6

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

   4. Applied rewrites 90.6%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.1

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

   9. Applied rewrites 68.1%

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

   11. Applied rewrites 88.9%

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

   if 4.0000000000000002e-115 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 59.9%

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

   3. Taylor expanded in t 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 78.5%

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

Alternative 11: 53.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 * y)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in t 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

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 52.7

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

5. Applied rewrites 52.7%

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

Developer Target 1: 99.7% 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 2024235 
(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))))