Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 50.1% → 80.2%

Time: 18.8s

Alternatives: 5

Speedup: 6.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-110}:\\ \;\;\;\;-1 + \frac{0.25}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-110)
     (+ -1.0 (/ 0.25 (/ (/ y x) (/ x y))))
     (if (<= (* x x) 5e+247)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-110) {
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	} else if ((x * x) <= 5e+247) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-110) then
        tmp = (-1.0d0) + (0.25d0 / ((y / x) / (x / y)))
    else if ((x * x) <= 5d+247) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-110) {
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	} else if ((x * x) <= 5e+247) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-110:
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)))
	elif (x * x) <= 5e+247:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-110)
		tmp = Float64(-1.0 + Float64(0.25 / Float64(Float64(y / x) / Float64(x / y))));
	elseif (Float64(x * x) <= 5e+247)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1e-110)
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	elseif ((x * x) <= 5e+247)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-110], N[(-1.0 + N[(0.25 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+247], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.0000000000000001e-110

   1. Initial program 55.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 47.7%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 47.7%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

   4. Simplified 47.7%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

   5. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 71.4%

         \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. *-commutative 71.4%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot 0.25} + \left(-1\right) \]

      3. associate-*l/ 71.4%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{{y}^{2}}} + \left(-1\right) \]

      4. associate-*r/ 71.4%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.25}{{y}^{2}}} + \left(-1\right) \]

      5. metadata-eval 71.4%

         \[\leadsto {x}^{2} \cdot \frac{0.25}{{y}^{2}} + \color{blue}{-1} \]

      6. +-commutative 71.4%

         \[\leadsto \color{blue}{-1 + {x}^{2} \cdot \frac{0.25}{{y}^{2}}} \]

      7. associate-*r/ 71.4%

         \[\leadsto -1 + \color{blue}{\frac{{x}^{2} \cdot 0.25}{{y}^{2}}} \]

      8. *-commutative 71.4%

         \[\leadsto -1 + \frac{\color{blue}{0.25 \cdot {x}^{2}}}{{y}^{2}} \]

      9. associate-/l* 71.4%

         \[\leadsto -1 + \color{blue}{\frac{0.25}{\frac{{y}^{2}}{{x}^{2}}}} \]

      10. unpow2 71.4%

          \[\leadsto -1 + \frac{0.25}{\frac{\color{blue}{y \cdot y}}{{x}^{2}}} \]

      11. unpow2 71.4%

          \[\leadsto -1 + \frac{0.25}{\frac{y \cdot y}{\color{blue}{x \cdot x}}} \]

      12. times-frac 79.3%

          \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]

      13. unpow2 79.3%

          \[\leadsto -1 + \frac{0.25}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   7. Simplified 79.3%

      \[\leadsto \color{blue}{-1 + \frac{0.25}{{\left(\frac{y}{x}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. pow2 79.3%

         \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]

      2. clear-num 79.3%

         \[\leadsto -1 + \frac{0.25}{\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}} \]

      3. un-div-inv 79.3%

         \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   9. Applied egg-rr 79.3%

      \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 1.0000000000000001e-110 < (*.f64 x x) < 5.00000000000000023e247

   1. Initial program 83.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 5.00000000000000023e247 < (*.f64 x x)

   1. Initial program 8.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 74.9%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 74.9%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 74.9%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 83.5%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   4. Applied egg-rr 83.5%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-110}:\\ \;\;\;\;-1 + \frac{0.25}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 2: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;-1 + \frac{0.25}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.7e+15) (+ -1.0 (/ 0.25 (/ (/ y x) (/ x y)))) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.7e+15) {
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.7d+15) then
        tmp = (-1.0d0) + (0.25d0 / ((y / x) / (x / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.7e+15) {
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.7e+15:
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.7e+15)
		tmp = Float64(-1.0 + Float64(0.25 / Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.7e+15)
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.7e+15], N[(-1.0 + N[(0.25 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.7e15

   1. Initial program 53.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 32.4%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 32.4%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

   4. Simplified 32.4%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 50.1%

         \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. *-commutative 50.1%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot 0.25} + \left(-1\right) \]

      3. associate-*l/ 50.1%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{{y}^{2}}} + \left(-1\right) \]

      4. associate-*r/ 50.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.25}{{y}^{2}}} + \left(-1\right) \]

      5. metadata-eval 50.1%

         \[\leadsto {x}^{2} \cdot \frac{0.25}{{y}^{2}} + \color{blue}{-1} \]

      6. +-commutative 50.1%

         \[\leadsto \color{blue}{-1 + {x}^{2} \cdot \frac{0.25}{{y}^{2}}} \]

      7. associate-*r/ 50.1%

         \[\leadsto -1 + \color{blue}{\frac{{x}^{2} \cdot 0.25}{{y}^{2}}} \]

      8. *-commutative 50.1%

         \[\leadsto -1 + \frac{\color{blue}{0.25 \cdot {x}^{2}}}{{y}^{2}} \]

      9. associate-/l* 50.1%

         \[\leadsto -1 + \color{blue}{\frac{0.25}{\frac{{y}^{2}}{{x}^{2}}}} \]

      10. unpow2 50.1%

          \[\leadsto -1 + \frac{0.25}{\frac{\color{blue}{y \cdot y}}{{x}^{2}}} \]

      11. unpow2 50.1%

          \[\leadsto -1 + \frac{0.25}{\frac{y \cdot y}{\color{blue}{x \cdot x}}} \]

      12. times-frac 56.5%

          \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]

      13. unpow2 56.5%

          \[\leadsto -1 + \frac{0.25}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{-1 + \frac{0.25}{{\left(\frac{y}{x}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. pow2 56.5%

         \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]

      2. clear-num 56.5%

         \[\leadsto -1 + \frac{0.25}{\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}} \]

      3. un-div-inv 56.5%

         \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   9. Applied egg-rr 56.5%

      \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 3.7e15 < x

   1. Initial program 30.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 71.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;-1 + \frac{0.25}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;-1 + \frac{0.25}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.4e+15)
   (+ -1.0 (/ 0.25 (/ (/ y x) (/ x y))))
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.4e+15) {
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 8.4d+15) then
        tmp = (-1.0d0) + (0.25d0 / ((y / x) / (x / y)))
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.4e+15) {
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.4e+15:
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)))
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.4e+15)
		tmp = Float64(-1.0 + Float64(0.25 / Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.4e+15)
		tmp = -1.0 + (0.25 / ((y / x) / (x / y)));
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.4e+15], N[(-1.0 + N[(0.25 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4e15

   1. Initial program 53.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 32.4%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 32.4%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

   4. Simplified 32.4%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 50.1%

         \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. *-commutative 50.1%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot 0.25} + \left(-1\right) \]

      3. associate-*l/ 50.1%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{{y}^{2}}} + \left(-1\right) \]

      4. associate-*r/ 50.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.25}{{y}^{2}}} + \left(-1\right) \]

      5. metadata-eval 50.1%

         \[\leadsto {x}^{2} \cdot \frac{0.25}{{y}^{2}} + \color{blue}{-1} \]

      6. +-commutative 50.1%

         \[\leadsto \color{blue}{-1 + {x}^{2} \cdot \frac{0.25}{{y}^{2}}} \]

      7. associate-*r/ 50.1%

         \[\leadsto -1 + \color{blue}{\frac{{x}^{2} \cdot 0.25}{{y}^{2}}} \]

      8. *-commutative 50.1%

         \[\leadsto -1 + \frac{\color{blue}{0.25 \cdot {x}^{2}}}{{y}^{2}} \]

      9. associate-/l* 50.1%

         \[\leadsto -1 + \color{blue}{\frac{0.25}{\frac{{y}^{2}}{{x}^{2}}}} \]

      10. unpow2 50.1%

          \[\leadsto -1 + \frac{0.25}{\frac{\color{blue}{y \cdot y}}{{x}^{2}}} \]

      11. unpow2 50.1%

          \[\leadsto -1 + \frac{0.25}{\frac{y \cdot y}{\color{blue}{x \cdot x}}} \]

      12. times-frac 56.5%

          \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]

      13. unpow2 56.5%

          \[\leadsto -1 + \frac{0.25}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{-1 + \frac{0.25}{{\left(\frac{y}{x}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. pow2 56.5%

         \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]

      2. clear-num 56.5%

         \[\leadsto -1 + \frac{0.25}{\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}} \]

      3. un-div-inv 56.5%

         \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   9. Applied egg-rr 56.5%

      \[\leadsto -1 + \frac{0.25}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 8.4e15 < x

   1. Initial program 30.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 67.6%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 67.6%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 67.6%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 72.4%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   4. Applied egg-rr 72.4%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+15}:\\ \;\;\;\;-1 + \frac{0.25}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 4: 62.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.35e+17) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.35e+17) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.35d+17) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.35e+17) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.35e+17:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.35e+17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.35e+17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.35e+17], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e17

   1. Initial program 53.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 55.1%

      \[\leadsto \color{blue}{-1} \]

   if 1.35e17 < x

   1. Initial program 30.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 71.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 50.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 47.2%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

2. Taylor expanded in x around 0 48.4%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 48.4%

   \[\leadsto -1 \]

Developer target: 50.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023313 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))