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

Percentage Accurate: 67.2% → 99.7%

Time: 9.2s

Alternatives: 12

Speedup: 0.8×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[Power[N[(x / y), $MachinePrecision], 2.0], $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. +-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 82.5

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

   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. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e+268)
     (+ (* (/ x y) (/ x y)) t_1)
     (if (<= t_1 INFINITY)
       (/ (/ z t) (/ t z))
       (fma (/ z t) (/ z 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+268) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = fma((z / t), (z / t), ((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+268)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x * x) / Float64(y * 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+268], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999999e268

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 1.9999999999999999e268 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 91.8%

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

   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. 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 87.7

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 87.7

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

   6. Applied rewrites 87.7%

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

Alternative 3: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x + t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e+268)
     (+ (* (/ (/ x y) y) x) t_1)
     (if (<= t_1 INFINITY)
       (/ (/ z t) (/ t z))
       (fma (/ z t) (/ z 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+268) {
		tmp = (((x / y) / y) * x) + t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = fma((z / t), (z / t), ((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+268)
		tmp = Float64(Float64(Float64(Float64(x / y) / y) * x) + t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x * x) / Float64(y * 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+268], N[(N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999999e268

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if 1.9999999999999999e268 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 91.8%

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

   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. 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 87.7

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 87.7

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

   6. Applied rewrites 87.7%

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

Alternative 4: 80.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (/ (/ z t) (/ t z))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 INFINITY) (+ t_1 (/ (* z z) (* t t))) 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 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 + ((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 = (x * x) / (y * y);
	double t_2 = (z / t) / (t / z);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 + ((z * z) / (t * t));
	} 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 <= 0.0:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = t_1 + ((z * z) / (t * t))
	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 / t) / Float64(t / z))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(t_1 + Float64(Float64(z * z) / Float64(t * t)));
	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 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_1 + ((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[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[(t$95$1 + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 0.0 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 53.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 84.0%

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

   if 0.0 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 78.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e+289)
     (+ (* (/ x y) (/ x y)) t_1)
     (fma (/ z t) (/ z 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+289) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = fma((z / t), (z / t), ((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+289)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	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_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+289], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $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)) < 2.0000000000000001e289

   1. Initial program 76.1%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 2.0000000000000001e289 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. +-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 84.5

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 97.4

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

   6. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 95.4

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

   8. Applied rewrites 95.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = fma(Float64(z / t), Float64(z / t), 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[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] * N[(z / t), $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 x x) (*.f64 y y)) < +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

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

      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 93.0

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 93.0

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

   6. Applied rewrites 93.0%

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

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

   1. Initial program 0.0%

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

   3. 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

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 53.5%

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

Alternative 7: 95.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.0

   1. Initial program 58.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. +-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 83.8

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 95.8

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

   8. Applied rewrites 95.8%

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

   if 0.0 < (*.f64 t t)

   1. Initial program 69.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 95.9%

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

Alternative 8: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y} \cdot x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{\frac{y}{x} \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.9e-240)
   (fma (/ z t) (/ z t) (/ (* (/ x y) x) y))
   (fma (/ z t) (/ z t) (/ x (* (/ y x) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.9e-240) {
		tmp = fma((z / t), (z / t), (((x / y) * x) / y));
	} else {
		tmp = fma((z / t), (z / t), (x / ((y / x) * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.9e-240)
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(Float64(x / y) * x) / y));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(x / Float64(Float64(y / x) * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.9e-240], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x / N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999994e-240

   1. Initial program 65.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 80.4

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 94.8

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

   6. Applied rewrites 94.8%

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

   if 1.89999999999999994e-240 < y

   1. Initial program 68.9%

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

      1. lift-+.f64 N/A

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

      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 84.9

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

      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. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 98.8

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

   6. Applied rewrites 98.8%

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

Alternative 9: 96.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x / N[(N[(y / x), $MachinePrecision] * 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. 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 82.5

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

   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. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 97.0

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

6. Applied rewrites 97.0%

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

Alternative 10: 57.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{z}{t}}{\frac{t}{z}} \end{array} \]

FPCore

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

C

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

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 = (z / t) / (t / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] / N[(t / z), $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 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. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 57.3

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

5. Applied rewrites 57.3%

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

7. Applied rewrites 59.8%

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

Alternative 11: 57.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (z / t) * (z / 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 = (z / t) * (z / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $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 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. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 57.3

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

5. Applied rewrites 57.3%

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

7. Applied rewrites 59.8%

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

Alternative 12: 51.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 = (z / (t * t)) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $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 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. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 57.3

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

5. Applied rewrites 57.3%

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

7. Applied rewrites 52.9%

   \[\leadsto \frac{z}{t \cdot t} \cdot z \]
8. 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 2024294 
(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))))