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

Percentage Accurate: 66.4% → 99.7%

Time: 10.2s

Alternatives: 10

Speedup: 1.1×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x / y) * (x / y)) + pow((t / z), -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) * (x / y)) + ((t / z) ** (-2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

   9. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

3. Simplified 81.3%

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

   1. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot \frac{x}{y}}{y}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(z, z\right)}, \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{y} \cdot \frac{x}{y}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(z, z\right)}, \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y}\right), \left(\frac{x}{y}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(z, z\right)}, \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\frac{x}{y}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{z}, z\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   5. /-lowering-/.f64 83.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{z}\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

6. Applied egg-rr 83.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \left(\frac{1}{\color{blue}{\frac{t \cdot t}{z \cdot z}}}\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \left({\left(\frac{t \cdot t}{z \cdot z}\right)}^{\color{blue}{-1}}\right)\right) \]

   3. times-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \left({\left(\frac{t}{z} \cdot \frac{t}{z}\right)}^{-1}\right)\right) \]

   4. unpow-prod-down N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \left({\left(\frac{t}{z}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{z}\right)}^{-1}}\right)\right) \]

   5. pow-prod-up N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \left({\left(\frac{t}{z}\right)}^{\color{blue}{\left(-1 + -1\right)}}\right)\right) \]

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{pow.f64}\left(\left(\frac{t}{z}\right), \color{blue}{\left(-1 + -1\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, z\right), \left(\color{blue}{-1} + -1\right)\right)\right) \]

   8. metadata-eval 99.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, z\right), -2\right)\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{{\left(\frac{t}{z}\right)}^{-2}} \]
9. Add Preprocessing

Alternative 2: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e-140) (/ (/ x y) (/ y x)) (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-140) {
		tmp = (x / y) / (y / x);
	} else {
		tmp = (z / t) * (z / t);
	}
	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 (((z * z) / (t * t)) <= 2d-140) then
        tmp = (x / y) / (y / x)
    else
        tmp = (z / t) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-140) {
		tmp = (x / y) / (y / x);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 2e-140:
		tmp = (x / y) / (y / x)
	else:
		tmp = (z / t) * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e-140)
		tmp = Float64(Float64(x / y) / Float64(y / x));
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 2e-140)
		tmp = (x / y) / (y / x);
	else
		tmp = (z / t) * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e-140], N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e-140

   1. Initial program 67.6%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 88.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   7. Simplified 69.2%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

      7. /-lowering-/.f64 89.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 89.1%

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

   if 2e-140 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 67.4%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{{t}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left({t}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(t \cdot \color{blue}{t}\right)\right)\right) \]

      6. *-lowering-*.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   7. Simplified 69.2%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{z}{t}\right), \color{blue}{\left(\frac{z}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \left(\frac{\color{blue}{z}}{t}\right)\right) \]

      5. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 78.9%

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

Alternative 3: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e-140) (* (/ x y) (/ x y)) (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-140) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	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 (((z * z) / (t * t)) <= 2d-140) then
        tmp = (x / y) * (x / y)
    else
        tmp = (z / t) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-140) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 2e-140:
		tmp = (x / y) * (x / y)
	else:
		tmp = (z / t) * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e-140)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 2e-140)
		tmp = (x / y) * (x / y);
	else
		tmp = (z / t) * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e-140], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e-140

   1. Initial program 67.6%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 88.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   7. Simplified 69.2%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      5. /-lowering-/.f64 89.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Applied egg-rr 89.0%

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

   if 2e-140 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 67.4%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{{t}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left({t}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(t \cdot \color{blue}{t}\right)\right)\right) \]

      6. *-lowering-*.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   7. Simplified 69.2%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{z}{t}\right), \color{blue}{\left(\frac{z}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \left(\frac{\color{blue}{z}}{t}\right)\right) \]

      5. /-lowering-/.f64 78.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 78.9%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

   9. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

3. Simplified 81.3%

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

   1. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot \frac{x}{y}}{y}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(z, z\right)}, \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{y} \cdot \frac{x}{y}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(z, z\right)}, \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y}\right), \left(\frac{x}{y}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(z, z\right)}, \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\frac{x}{y}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{z}, z\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   5. /-lowering-/.f64 83.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{z}\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

6. Applied egg-rr 83.9%

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

   1. times-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \left(\frac{z}{t} \cdot \color{blue}{\frac{z}{t}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\left(\frac{z}{t}\right), \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \left(\frac{\color{blue}{z}}{t}\right)\right)\right) \]

   4. /-lowering-/.f64 99.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, y\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

8. Applied egg-rr 99.6%

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

Alternative 5: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

   9. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

3. Simplified 81.3%

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

   1. times-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z}{t} \cdot \color{blue}{\frac{z}{t}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z}{t} \cdot \frac{1}{\color{blue}{\frac{t}{z}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(\frac{z}{t}\right), \color{blue}{\left(\frac{t}{z}\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(z, t\right), \left(\frac{\color{blue}{t}}{z}\right)\right)\right) \]

   6. /-lowering-/.f64 96.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(z, t\right), \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

6. Applied egg-rr 96.6%

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

Alternative 6: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

   9. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

3. Simplified 81.3%

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

   1. times-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z}{t} \cdot \color{blue}{\frac{z}{t}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{*.f64}\left(\left(\frac{z}{t}\right), \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \left(\frac{\color{blue}{z}}{t}\right)\right)\right) \]

   4. /-lowering-/.f64 96.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, t\right), \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

6. Applied egg-rr 96.5%

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

7. Final simplification 96.5%

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

Alternative 7: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{+69}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 1e+69) (* z (/ (/ z t) t)) (* (/ x y) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 1e+69) {
		tmp = z * ((z / t) / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t * t) <= 1d+69) then
        tmp = z * ((z / t) / t)
    else
        tmp = (x / y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 1e+69) {
		tmp = z * ((z / t) / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t * t) <= 1e+69:
		tmp = z * ((z / t) / t)
	else:
		tmp = (x / y) * (x / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 1e+69)
		tmp = Float64(z * Float64(Float64(z / t) / t));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * t) <= 1e+69)
		tmp = z * ((z / t) / t);
	else
		tmp = (x / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 1e+69], N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.0000000000000001e69

   1. Initial program 76.4%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 86.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{{t}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left({t}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(t \cdot \color{blue}{t}\right)\right)\right) \]

      6. *-lowering-*.f64 74.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   7. Simplified 74.5%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{z}{t}}{\color{blue}{t}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{z}{t}\right), \color{blue}{t}\right)\right) \]

      3. /-lowering-/.f64 77.4%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{/.f64}\left(z, t\right), t\right)\right) \]

   9. Applied egg-rr 77.4%

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

   if 1.0000000000000001e69 < (*.f64 t t)

   1. Initial program 57.1%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 75.5%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 61.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   7. Simplified 61.8%

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

      1. associate-*r/ N/A

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

      2. frac-times N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      5. /-lowering-/.f64 78.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   9. Applied egg-rr 78.6%

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

Alternative 8: 67.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 1.16 \cdot 10^{+75}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 1.16e+75) (* z (/ z (* t t))) (* x (/ x (* y y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 1.16e+75) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x * (x / (y * 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 ((t * t) <= 1.16d+75) then
        tmp = z * (z / (t * t))
    else
        tmp = x * (x / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 1.16e+75) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x * (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t * t) <= 1.16e+75:
		tmp = z * (z / (t * t))
	else:
		tmp = x * (x / (y * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 1.16e+75)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	else
		tmp = Float64(x * Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * t) <= 1.16e+75)
		tmp = z * (z / (t * t));
	else
		tmp = x * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 1.16e+75], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.1600000000000001e75

   1. Initial program 76.4%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 86.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{{t}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left({t}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(t \cdot \color{blue}{t}\right)\right)\right) \]

      6. *-lowering-*.f64 74.5%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   7. Simplified 74.5%

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

   if 1.1600000000000001e75 < (*.f64 t t)

   1. Initial program 57.1%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 75.5%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 61.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   7. Simplified 61.8%

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

Alternative 9: 62.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+30}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 6.2e+30) (* z (/ (/ z t) t)) (* x (/ x (* y y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6.2e+30) {
		tmp = z * ((z / t) / t);
	} else {
		tmp = x * (x / (y * 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 <= 6.2d+30) then
        tmp = z * ((z / t) / t)
    else
        tmp = x * (x / (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6.2e+30) {
		tmp = z * ((z / t) / t);
	} else {
		tmp = x * (x / (y * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 6.2e+30:
		tmp = z * ((z / t) / t)
	else:
		tmp = x * (x / (y * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 6.2e+30)
		tmp = Float64(z * Float64(Float64(z / t) / t));
	else
		tmp = Float64(x * Float64(x / Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 6.2e+30)
		tmp = z * ((z / t) / t);
	else
		tmp = x * (x / (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 6.2e+30], N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.1999999999999995e30

   1. Initial program 67.9%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 80.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{{t}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left({t}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(t \cdot \color{blue}{t}\right)\right)\right) \]

      6. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   7. Simplified 56.8%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{\frac{z}{t}}{\color{blue}{t}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{z}{t}\right), \color{blue}{t}\right)\right) \]

      3. /-lowering-/.f64 63.4%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(\mathsf{/.f64}\left(z, t\right), t\right)\right) \]

   9. Applied egg-rr 63.4%

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

   if 6.1999999999999995e30 < x

   1. Initial program 66.2%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

      9. *-lowering-*.f64 85.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   3. Simplified 85.2%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   7. Simplified 74.4%

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

Alternative 10: 52.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot x}{y \cdot y}\right), \color{blue}{\left(\frac{z \cdot z}{t \cdot t}\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{x}{y \cdot y}\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{x}{y \cdot y}\right)\right), \left(\frac{\color{blue}{z \cdot z}}{t \cdot t}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{x}{y}}{y}\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), y\right)\right), \left(\frac{z \cdot \color{blue}{z}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \left(\frac{z \cdot z}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{t} \cdot t\right)\right)\right) \]

   9. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

3. Simplified 81.3%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{x}{{y}^{2}}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   6. *-lowering-*.f64 52.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

7. Simplified 52.9%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024138 
(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))))