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

Percentage Accurate: 66.4% → 99.7%

Time: 16.1s

Alternatives: 8

Speedup: 1.0×

• 48.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. times-frac 83.4%

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

3. Simplified 83.4%

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

   1. frac-times 99.7%

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

5. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-170}:\\ \;\;\;\;\frac{t}{y \cdot t} \cdot \frac{x \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e-170)
   (* (/ t (* y t)) (/ (* x x) y))
   (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e-170) {
		tmp = (t / (y * t)) * ((x * 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)) <= 5d-170) then
        tmp = (t / (y * t)) * ((x * 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)) <= 5e-170) {
		tmp = (t / (y * t)) * ((x * x) / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 5e-170:
		tmp = (t / (y * t)) * ((x * 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)) <= 5e-170)
		tmp = Float64(Float64(t / Float64(y * t)) * Float64(Float64(x * 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)) <= 5e-170)
		tmp = (t / (y * t)) * ((x * 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], 5e-170], N[(N[(t / N[(y * t), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / 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)) < 5.0000000000000001e-170

   1. Initial program 76.2%

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

      1. associate-*l/ 83.4%

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

      2. *-commutative 83.4%

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

      3. fma-def 83.4%

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

   3. Simplified 83.4%

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

      1. fma-udef 83.4%

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

      2. associate-*r/ 76.2%

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

      3. frac-times 98.5%

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

      4. frac-times 99.6%

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

      5. +-commutative 99.6%

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

      6. frac-times 98.5%

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

      7. associate-/r* 99.6%

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

      8. associate-*l/ 94.0%

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

      9. frac-add 79.8%

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

      10. fma-def 79.8%

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

      11. associate-*r/ 79.8%

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

   5. Applied egg-rr 79.8%

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

      1. associate-*r* 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*l* 81.0%

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

      4. *-commutative 81.0%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in z around 0 70.4%

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

      1. unpow2 70.4%

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

      2. associate-*r/ 70.6%

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

      3. associate-*l/ 76.4%

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

   10. Simplified 76.4%

       \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)}}{y \cdot t} \]
   11. Step-by-step derivation

       1. associate-/l* 82.4%

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

       2. associate-/r/ 83.5%

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

       3. associate-*l/ 75.3%

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

   12. Applied egg-rr 75.3%

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

   if 5.0000000000000001e-170 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.8%

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

      1. associate-*l/ 74.4%

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

      2. *-commutative 74.4%

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

      3. fma-def 74.4%

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

   3. Simplified 74.4%

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

      1. fma-udef 74.4%

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

      2. associate-*r/ 65.8%

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

      3. frac-times 75.0%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. frac-times 75.0%

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

      7. clear-num 74.9%

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

      8. frac-times 65.8%

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

      9. associate-/l* 74.4%

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

      10. frac-add 36.5%

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

      11. *-un-lft-identity 36.5%

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

      12. associate-/l* 38.8%

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

      13. times-frac 40.5%

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

      14. times-frac 52.3%

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

      15. associate-/l* 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. associate-/l* 53.4%

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

      2. associate-*r/ 60.3%

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

      3. *-commutative 60.3%

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

      4. associate-*r/ 59.7%

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

      5. associate-/r/ 59.7%

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

      6. associate-*r/ 58.7%

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

      7. associate-/r/ 58.6%

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

      8. associate-/l* 51.1%

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

      9. associate-*r/ 58.7%

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

   7. Simplified 58.7%

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

   8. Taylor expanded in y around inf 69.0%

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

      1. unpow2 69.0%

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

      2. unpow2 69.0%

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

   10. Simplified 69.0%

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

       1. frac-times 81.3%

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

   12. Applied egg-rr 81.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-170}:\\ \;\;\;\;\frac{t}{y \cdot t} \cdot \frac{x \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]

Alternative 3: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\frac{x}{y} \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e-170)
   (* (/ x y) (/ (* (/ x y) t) t))
   (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e-170) {
		tmp = (x / y) * (((x / y) * t) / t);
	} 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)) <= 5d-170) then
        tmp = (x / y) * (((x / y) * t) / t)
    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)) <= 5e-170) {
		tmp = (x / y) * (((x / y) * t) / t);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 5e-170:
		tmp = (x / y) * (((x / y) * t) / t)
	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)) <= 5e-170)
		tmp = Float64(Float64(x / y) * Float64(Float64(Float64(x / y) * t) / t));
	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)) <= 5e-170)
		tmp = (x / y) * (((x / y) * t) / t);
	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], 5e-170], N[(N[(x / y), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] * t), $MachinePrecision] / t), $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)) < 5.0000000000000001e-170

   1. Initial program 76.2%

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

      1. associate-*l/ 83.4%

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

      2. *-commutative 83.4%

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

      3. fma-def 83.4%

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

   3. Simplified 83.4%

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

      1. fma-udef 83.4%

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

      2. associate-*r/ 76.2%

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

      3. frac-times 98.5%

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

      4. frac-times 99.6%

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

      5. +-commutative 99.6%

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

      6. frac-times 98.5%

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

      7. associate-/r* 99.6%

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

      8. associate-*l/ 94.0%

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

      9. frac-add 79.8%

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

      10. fma-def 79.8%

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

      11. associate-*r/ 79.8%

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

   5. Applied egg-rr 79.8%

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

      1. associate-*r* 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*l* 81.0%

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

      4. *-commutative 81.0%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in z around 0 70.4%

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

      1. unpow2 70.4%

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

      2. associate-*r/ 70.6%

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

      3. associate-*l/ 76.4%

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

   10. Simplified 76.4%

       \[\leadsto \frac{\color{blue}{t \cdot \left(\frac{x}{y} \cdot x\right)}}{y \cdot t} \]
   11. Step-by-step derivation

       1. associate-*r* 77.6%

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

       2. *-commutative 77.6%

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

       3. times-frac 90.5%

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

       4. *-commutative 90.5%

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

   12. Applied egg-rr 90.5%

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

   if 5.0000000000000001e-170 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.8%

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

      1. associate-*l/ 74.4%

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

      2. *-commutative 74.4%

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

      3. fma-def 74.4%

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

   3. Simplified 74.4%

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

      1. fma-udef 74.4%

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

      2. associate-*r/ 65.8%

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

      3. frac-times 75.0%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. frac-times 75.0%

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

      7. clear-num 74.9%

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

      8. frac-times 65.8%

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

      9. associate-/l* 74.4%

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

      10. frac-add 36.5%

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

      11. *-un-lft-identity 36.5%

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

      12. associate-/l* 38.8%

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

      13. times-frac 40.5%

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

      14. times-frac 52.3%

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

      15. associate-/l* 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. associate-/l* 53.4%

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

      2. associate-*r/ 60.3%

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

      3. *-commutative 60.3%

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

      4. associate-*r/ 59.7%

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

      5. associate-/r/ 59.7%

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

      6. associate-*r/ 58.7%

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

      7. associate-/r/ 58.6%

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

      8. associate-/l* 51.1%

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

      9. associate-*r/ 58.7%

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

   7. Simplified 58.7%

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

   8. Taylor expanded in y around inf 69.0%

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

      1. unpow2 69.0%

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

      2. unpow2 69.0%

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

   10. Simplified 69.0%

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

       1. frac-times 81.3%

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

   12. Applied egg-rr 81.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\frac{x}{y} \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]

Alternative 4: 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 69.1%

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

   1. times-frac 83.4%

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

3. Simplified 83.4%

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

   1. frac-times 99.7%

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

5. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0 83.4%

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

   1. unpow2 83.4%

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

   2. unpow2 83.4%

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

   3. associate-*r/ 92.7%

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

   4. associate-/r* 98.3%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. times-frac 83.4%

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

3. Simplified 83.4%

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

   1. frac-times 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 59.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{1}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.9e+94) (* (/ z t) (/ z t)) (* z (* z (/ 1.0 (* t t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.9e+94) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = z * (z * (1.0 / (t * 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 (x <= 2.9d+94) then
        tmp = (z / t) * (z / t)
    else
        tmp = z * (z * (1.0d0 / (t * t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.9e+94) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = z * (z * (1.0 / (t * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.9e+94:
		tmp = (z / t) * (z / t)
	else:
		tmp = z * (z * (1.0 / (t * t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.9e+94)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(z * Float64(z * Float64(1.0 / Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.9e+94)
		tmp = (z / t) * (z / t);
	else
		tmp = z * (z * (1.0 / (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.9e+94], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(1.0 / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8999999999999998e94

   1. Initial program 71.8%

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

      1. associate-*l/ 79.7%

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

      2. *-commutative 79.7%

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

      3. fma-def 79.7%

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

   3. Simplified 79.7%

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

      1. fma-udef 79.7%

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

      2. associate-*r/ 71.8%

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

      3. frac-times 83.3%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. frac-times 83.3%

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

      7. clear-num 83.2%

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

      8. frac-times 71.8%

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

      9. associate-/l* 79.7%

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

      10. frac-add 27.8%

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

      11. *-un-lft-identity 27.8%

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

      12. associate-/l* 29.7%

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

      13. times-frac 30.6%

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

      14. times-frac 39.7%

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

      15. associate-/l* 44.0%

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

   5. Applied egg-rr 44.0%

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

      1. associate-/l* 39.6%

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

      2. associate-*r/ 43.9%

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

      3. *-commutative 43.9%

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

      4. associate-*r/ 43.9%

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

      5. associate-/r/ 43.9%

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

      6. associate-*r/ 42.6%

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

      7. associate-/r/ 42.6%

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

      8. associate-/l* 38.3%

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

      9. associate-*r/ 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y around inf 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

   10. Simplified 55.7%

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

       1. frac-times 66.1%

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

   12. Applied egg-rr 66.1%

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

   if 2.8999999999999998e94 < x

   1. Initial program 55.4%

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

      1. associate-*l/ 65.1%

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

      2. *-commutative 65.1%

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

      3. fma-def 65.1%

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

   3. Simplified 65.1%

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

      1. fma-udef 65.1%

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

      2. associate-*r/ 55.4%

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

      3. frac-times 78.6%

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

      4. frac-times 99.6%

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

      5. +-commutative 99.6%

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

      6. frac-times 78.6%

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

      7. clear-num 78.6%

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

      8. frac-times 55.4%

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

      9. associate-/l* 65.0%

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

      10. frac-add 23.9%

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

      11. *-un-lft-identity 23.9%

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

      12. associate-/l* 24.0%

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

      13. times-frac 26.4%

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

      14. times-frac 48.0%

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

      15. associate-/l* 59.6%

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

   5. Applied egg-rr 59.6%

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

      1. associate-/l* 52.8%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-*r/ 57.1%

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

      5. associate-/r/ 57.1%

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

      6. associate-*r/ 54.9%

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

      7. associate-/r/ 55.0%

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

      8. associate-/l* 45.6%

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

      9. associate-*r/ 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in y around inf 32.6%

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

      1. unpow2 32.6%

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

      2. unpow2 32.6%

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

   10. Simplified 32.6%

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

       1. associate-/l* 40.1%

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

       2. div-inv 40.1%

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

       3. associate-/l* 33.5%

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

   12. Applied egg-rr 33.5%

       \[\leadsto \color{blue}{z \cdot \frac{1}{\frac{t}{\frac{z}{t}}}} \]
   13. Step-by-step derivation

       1. associate-/r/ 33.4%

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

       2. associate-*l/ 40.1%

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

       3. associate-/r/ 40.1%

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

   14. Applied egg-rr 40.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{1}{t \cdot t}\right)\\ \end{array} \]

Alternative 7: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 5e+93) (* (/ z t) (/ z t)) (* z (/ z (* t t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5e+93) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = z * (z / (t * 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 (x <= 5d+93) then
        tmp = (z / t) * (z / t)
    else
        tmp = z * (z / (t * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5e+93) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = z * (z / (t * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 5e+93:
		tmp = (z / t) * (z / t)
	else:
		tmp = z * (z / (t * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 5e+93)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(z * Float64(z / Float64(t * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 5e+93)
		tmp = (z / t) * (z / t);
	else
		tmp = z * (z / (t * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 5e+93], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e93

   1. Initial program 71.8%

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

      1. associate-*l/ 79.7%

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

      2. *-commutative 79.7%

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

      3. fma-def 79.7%

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

   3. Simplified 79.7%

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

      1. fma-udef 79.7%

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

      2. associate-*r/ 71.8%

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

      3. frac-times 83.3%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. frac-times 83.3%

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

      7. clear-num 83.2%

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

      8. frac-times 71.8%

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

      9. associate-/l* 79.7%

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

      10. frac-add 27.8%

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

      11. *-un-lft-identity 27.8%

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

      12. associate-/l* 29.7%

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

      13. times-frac 30.6%

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

      14. times-frac 39.7%

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

      15. associate-/l* 44.0%

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

   5. Applied egg-rr 44.0%

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

      1. associate-/l* 39.6%

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

      2. associate-*r/ 43.9%

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

      3. *-commutative 43.9%

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

      4. associate-*r/ 43.9%

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

      5. associate-/r/ 43.9%

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

      6. associate-*r/ 42.6%

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

      7. associate-/r/ 42.6%

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

      8. associate-/l* 38.3%

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

      9. associate-*r/ 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y around inf 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

   10. Simplified 55.7%

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

       1. frac-times 66.1%

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

   12. Applied egg-rr 66.1%

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

   if 5.0000000000000001e93 < x

   1. Initial program 55.4%

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

      1. associate-*l/ 65.1%

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

      2. *-commutative 65.1%

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

      3. fma-def 65.1%

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

   3. Simplified 65.1%

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

      1. fma-udef 65.1%

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

      2. associate-*r/ 55.4%

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

      3. frac-times 78.6%

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

      4. frac-times 99.6%

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

      5. +-commutative 99.6%

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

      6. frac-times 78.6%

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

      7. clear-num 78.6%

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

      8. frac-times 55.4%

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

      9. associate-/l* 65.0%

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

      10. frac-add 23.9%

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

      11. *-un-lft-identity 23.9%

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

      12. associate-/l* 24.0%

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

      13. times-frac 26.4%

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

      14. times-frac 48.0%

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

      15. associate-/l* 59.6%

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

   5. Applied egg-rr 59.6%

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

      1. associate-/l* 52.8%

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

      2. associate-*r/ 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-*r/ 57.1%

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

      5. associate-/r/ 57.1%

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

      6. associate-*r/ 54.9%

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

      7. associate-/r/ 55.0%

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

      8. associate-/l* 45.6%

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

      9. associate-*r/ 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in y around inf 32.6%

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

      1. unpow2 32.6%

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

      2. unpow2 32.6%

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

   10. Simplified 32.6%

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

       1. associate-/l* 40.1%

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

       2. associate-/r/ 40.1%

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

   12. Applied egg-rr 40.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \end{array} \]

Alternative 8: 58.9% accurate, 2.1× 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 69.1%

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

   1. associate-*l/ 77.3%

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

   2. *-commutative 77.3%

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

   3. fma-def 77.3%

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

3. Simplified 77.3%

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

   1. fma-udef 77.3%

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

   2. associate-*r/ 69.1%

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

   3. frac-times 82.5%

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

   4. frac-times 99.7%

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

   5. +-commutative 99.7%

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

   6. frac-times 82.5%

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

   7. clear-num 82.5%

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

   8. frac-times 69.1%

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

   9. associate-/l* 77.3%

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

   10. frac-add 27.2%

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

   11. *-un-lft-identity 27.2%

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

   12. associate-/l* 28.8%

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

   13. times-frac 30.0%

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

   14. times-frac 41.0%

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

   15. associate-/l* 46.5%

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

5. Applied egg-rr 46.5%

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

   1. associate-/l* 41.8%

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

   2. associate-*r/ 46.5%

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

   3. *-commutative 46.5%

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

   4. associate-*r/ 46.1%

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

   5. associate-/r/ 46.1%

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

   6. associate-*r/ 44.6%

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

   7. associate-/r/ 44.6%

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

   8. associate-/l* 39.5%

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

   9. associate-*r/ 44.6%

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

7. Simplified 44.6%

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

8. Taylor expanded in y around inf 51.9%

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

   1. unpow2 51.9%

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

   2. unpow2 51.9%

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

10. Simplified 51.9%

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

    1. frac-times 60.7%

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

12. Applied egg-rr 60.7%

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

13. Final simplification 60.7%

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

Developer target: 99.6% 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 2023279 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

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