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

Percentage Accurate: 50.9% → 81.6%

Time: 12.1s

Alternatives: 9

Speedup: 1.2×

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: 50.9% 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: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     (if (<= t_0 2e+267)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ -1.0 (* 0.5 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 2e+267) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	}
	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 (t_0 <= 0.0d0) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 2d+267) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = (-1.0d0) + (0.5d0 * ((x / y) * (x / y)))
    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 (t_0 <= 0.0) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 2e+267) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	elif t_0 <= 2e+267:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 2e+267)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 2e+267)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+267], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 0.0

   1. Initial program 56.7%

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

   3. Taylor expanded in y around 0 68.3%

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

      1. unpow2 68.3%

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

      2. pow2 68.3%

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

      3. times-frac 83.0%

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

   5. Applied egg-rr 83.0%

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

   if 0.0 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.9999999999999999e267

   1. Initial program 82.5%

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

   if 1.9999999999999999e267 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0 80.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. pow2 80.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 80.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 86.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

   5. Applied egg-rr 86.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x \cdot x - t\_0\\ \mathbf{if}\;\frac{t\_1}{x \cdot x + t\_0} \leq 2:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (- (* x x) t_0)))
   (if (<= (/ t_1 (+ (* x x) t_0)) 2.0)
     (/ t_1 (fma x x (* 4.0 (pow y 2.0))))
     (+ (log (hypot 1.0 (/ x y))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x * x) - t_0;
	double tmp;
	if ((t_1 / ((x * x) + t_0)) <= 2.0) {
		tmp = t_1 / fma(x, x, (4.0 * pow(y, 2.0)));
	} else {
		tmp = log(hypot(1.0, (x / y))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x * x) - t_0)
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(x * x) + t_0)) <= 2.0)
		tmp = Float64(t_1 / fma(x, x, Float64(4.0 * (y ^ 2.0))));
	else
		tmp = Float64(log(hypot(1.0, Float64(x / y))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$1 / N[(x * x + N[(4.0 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Sqrt[1.0 ^ 2 + N[(x / y), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 99.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-define 99.6%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot y\right)} \]

      3. associate-*l* 99.7%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot y\right)}\right)} \]

      4. pow2 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 47.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. pow2 47.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. add-log-exp 47.2%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{\frac{x \cdot x}{{y}^{2}}}\right)} - 1 \]

      3. add-sqr-sqrt 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{x \cdot x}{{y}^{2}}} \cdot \sqrt{\frac{x \cdot x}{{y}^{2}}}}}\right) - 1 \]

      4. pow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{{\left(\sqrt{\frac{x \cdot x}{{y}^{2}}}\right)}^{2}}}\right) - 1 \]

      5. sqrt-div 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{{\color{blue}{\left(\frac{\sqrt{x \cdot x}}{\sqrt{{y}^{2}}}\right)}}^{2}}\right) - 1 \]

      6. sqrt-prod 22.3%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      7. add-sqr-sqrt 50.5%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{x}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      8. sqrt-pow1 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{{y}^{\left(\frac{2}{2}\right)}}}\right)}^{2}}\right) - 1 \]

      9. metadata-eval 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{{y}^{\color{blue}{1}}}\right)}^{2}}\right) - 1 \]

      10. pow1 60.0%

          \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{y}}\right)}^{2}}\right) - 1 \]

   5. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{{\left(\frac{x}{y}\right)}^{2}}\right)} - 1 \]

   6. Taylor expanded in x around 0 47.2%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 + \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   7. Step-by-step derivation

      1. +-commutative 47.2%

         \[\leadsto 0.5 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

      2. unpow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} + 1\right) - 1 \]

      3. associate-*r/ 51.4%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} + 1\right) - 1 \]

      4. unpow2 51.4%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \frac{x}{\color{blue}{y \cdot y}} + 1\right) - 1 \]

      5. associate-/l/ 62.0%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \color{blue}{\frac{\frac{x}{y}}{y}} + 1\right) - 1 \]

      6. associate-*r/ 61.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x \cdot \frac{x}{y}}{y}} + 1\right) - 1 \]

      7. associate-*l/ 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + 1\right) - 1 \]

      8. unpow2 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + 1\right) - 1 \]

   8. Simplified 62.5%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + 1\right)} - 1 \]
   9. Step-by-step derivation

      1. sub-neg 62.5%

         \[\leadsto \color{blue}{0.5 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} + 1\right) + \left(-1\right)} \]

      2. add-log-exp 62.4%

         \[\leadsto \color{blue}{\log \left(e^{0.5 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} + 1\right)}\right)} + \left(-1\right) \]

      3. *-commutative 62.4%

         \[\leadsto \log \left(e^{\color{blue}{\log \left({\left(\frac{x}{y}\right)}^{2} + 1\right) \cdot 0.5}}\right) + \left(-1\right) \]

      4. exp-to-pow 62.4%

         \[\leadsto \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{2} + 1\right)}^{0.5}\right)} + \left(-1\right) \]

      5. pow1/2 62.4%

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

      6. +-commutative 62.4%

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

      7. unpow2 62.4%

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

      8. hypot-1-def 63.3%

         \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right)} + \left(-1\right) \]

      9. metadata-eval 63.3%

         \[\leadsto \log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + \color{blue}{-1} \]

   10. Applied egg-rr 63.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)} \leq 2:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x \cdot x - t\_0\\ \mathbf{if}\;\frac{t\_1}{x \cdot x + t\_0} \leq 2:\\ \;\;\;\;\frac{t\_1}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (- (* x x) t_0)))
   (if (<= (/ t_1 (+ (* x x) t_0)) 2.0)
     (/ t_1 (pow (hypot x (* y 2.0)) 2.0))
     (+ (log (hypot 1.0 (/ x y))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x * x) - t_0;
	double tmp;
	if ((t_1 / ((x * x) + t_0)) <= 2.0) {
		tmp = t_1 / pow(hypot(x, (y * 2.0)), 2.0);
	} else {
		tmp = log(hypot(1.0, (x / y))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x * x) - t_0;
	double tmp;
	if ((t_1 / ((x * x) + t_0)) <= 2.0) {
		tmp = t_1 / Math.pow(Math.hypot(x, (y * 2.0)), 2.0);
	} else {
		tmp = Math.log(Math.hypot(1.0, (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = (x * x) - t_0
	tmp = 0
	if (t_1 / ((x * x) + t_0)) <= 2.0:
		tmp = t_1 / math.pow(math.hypot(x, (y * 2.0)), 2.0)
	else:
		tmp = math.log(math.hypot(1.0, (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x * x) - t_0)
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(x * x) + t_0)) <= 2.0)
		tmp = Float64(t_1 / (hypot(x, Float64(y * 2.0)) ^ 2.0));
	else
		tmp = Float64(log(hypot(1.0, Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = (x * x) - t_0;
	tmp = 0.0;
	if ((t_1 / ((x * x) + t_0)) <= 2.0)
		tmp = t_1 / (hypot(x, (y * 2.0)) ^ 2.0);
	else
		tmp = log(hypot(1.0, (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$1 / N[Power[N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Sqrt[1.0 ^ 2 + N[(x / y), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 99.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-define 99.6%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot y\right)} \cdot y\right)} \]

      3. associate-*l* 99.7%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \color{blue}{4 \cdot \left(y \cdot y\right)}\right)} \]

      4. pow2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. add-sqr-sqrt 99.6%

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

      2. pow2 99.6%

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

      3. fma-undefine 99.6%

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

      4. add-sqr-sqrt 99.6%

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

      5. hypot-define 99.6%

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

      6. *-commutative 99.6%

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

      7. sqrt-prod 99.6%

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

      8. sqrt-pow1 99.6%

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

      9. metadata-eval 99.6%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{{\left(\mathsf{hypot}\left(x, {y}^{\color{blue}{1}} \cdot \sqrt{4}\right)\right)}^{2}} \]

      10. pow1 99.6%

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

      11. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 47.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. pow2 47.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. add-log-exp 47.2%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{\frac{x \cdot x}{{y}^{2}}}\right)} - 1 \]

      3. add-sqr-sqrt 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{x \cdot x}{{y}^{2}}} \cdot \sqrt{\frac{x \cdot x}{{y}^{2}}}}}\right) - 1 \]

      4. pow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{{\left(\sqrt{\frac{x \cdot x}{{y}^{2}}}\right)}^{2}}}\right) - 1 \]

      5. sqrt-div 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{{\color{blue}{\left(\frac{\sqrt{x \cdot x}}{\sqrt{{y}^{2}}}\right)}}^{2}}\right) - 1 \]

      6. sqrt-prod 22.3%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      7. add-sqr-sqrt 50.5%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{x}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      8. sqrt-pow1 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{{y}^{\left(\frac{2}{2}\right)}}}\right)}^{2}}\right) - 1 \]

      9. metadata-eval 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{{y}^{\color{blue}{1}}}\right)}^{2}}\right) - 1 \]

      10. pow1 60.0%

          \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{y}}\right)}^{2}}\right) - 1 \]

   5. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{{\left(\frac{x}{y}\right)}^{2}}\right)} - 1 \]

   6. Taylor expanded in x around 0 47.2%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 + \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   7. Step-by-step derivation

      1. +-commutative 47.2%

         \[\leadsto 0.5 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

      2. unpow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} + 1\right) - 1 \]

      3. associate-*r/ 51.4%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} + 1\right) - 1 \]

      4. unpow2 51.4%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \frac{x}{\color{blue}{y \cdot y}} + 1\right) - 1 \]

      5. associate-/l/ 62.0%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \color{blue}{\frac{\frac{x}{y}}{y}} + 1\right) - 1 \]

      6. associate-*r/ 61.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x \cdot \frac{x}{y}}{y}} + 1\right) - 1 \]

      7. associate-*l/ 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + 1\right) - 1 \]

      8. unpow2 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + 1\right) - 1 \]

   8. Simplified 62.5%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + 1\right)} - 1 \]
   9. Step-by-step derivation

      1. sub-neg 62.5%

         \[\leadsto \color{blue}{0.5 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} + 1\right) + \left(-1\right)} \]

      2. add-log-exp 62.4%

         \[\leadsto \color{blue}{\log \left(e^{0.5 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} + 1\right)}\right)} + \left(-1\right) \]

      3. *-commutative 62.4%

         \[\leadsto \log \left(e^{\color{blue}{\log \left({\left(\frac{x}{y}\right)}^{2} + 1\right) \cdot 0.5}}\right) + \left(-1\right) \]

      4. exp-to-pow 62.4%

         \[\leadsto \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{2} + 1\right)}^{0.5}\right)} + \left(-1\right) \]

      5. pow1/2 62.4%

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

      6. +-commutative 62.4%

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

      7. unpow2 62.4%

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

      8. hypot-1-def 63.3%

         \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right)} + \left(-1\right) \]

      9. metadata-eval 63.3%

         \[\leadsto \log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + \color{blue}{-1} \]

   10. Applied egg-rr 63.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)} \leq 2:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 2.0) t_1 (+ (log (hypot 1.0 (/ x y))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = log(hypot(1.0, (x / y))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = Math.log(Math.hypot(1.0, (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_1
	else:
		tmp = math.log(math.hypot(1.0, (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(log(hypot(1.0, Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = log(hypot(1.0, (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2.0], t$95$1, N[(N[Log[N[Sqrt[1.0 ^ 2 + N[(x / y), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 99.6%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 47.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. pow2 47.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. add-log-exp 47.2%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{\frac{x \cdot x}{{y}^{2}}}\right)} - 1 \]

      3. add-sqr-sqrt 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{x \cdot x}{{y}^{2}}} \cdot \sqrt{\frac{x \cdot x}{{y}^{2}}}}}\right) - 1 \]

      4. pow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{{\left(\sqrt{\frac{x \cdot x}{{y}^{2}}}\right)}^{2}}}\right) - 1 \]

      5. sqrt-div 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{{\color{blue}{\left(\frac{\sqrt{x \cdot x}}{\sqrt{{y}^{2}}}\right)}}^{2}}\right) - 1 \]

      6. sqrt-prod 22.3%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      7. add-sqr-sqrt 50.5%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{x}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      8. sqrt-pow1 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{{y}^{\left(\frac{2}{2}\right)}}}\right)}^{2}}\right) - 1 \]

      9. metadata-eval 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{{y}^{\color{blue}{1}}}\right)}^{2}}\right) - 1 \]

      10. pow1 60.0%

          \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{y}}\right)}^{2}}\right) - 1 \]

   5. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{{\left(\frac{x}{y}\right)}^{2}}\right)} - 1 \]

   6. Taylor expanded in x around 0 47.2%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 + \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   7. Step-by-step derivation

      1. +-commutative 47.2%

         \[\leadsto 0.5 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

      2. unpow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} + 1\right) - 1 \]

      3. associate-*r/ 51.4%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} + 1\right) - 1 \]

      4. unpow2 51.4%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \frac{x}{\color{blue}{y \cdot y}} + 1\right) - 1 \]

      5. associate-/l/ 62.0%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \color{blue}{\frac{\frac{x}{y}}{y}} + 1\right) - 1 \]

      6. associate-*r/ 61.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x \cdot \frac{x}{y}}{y}} + 1\right) - 1 \]

      7. associate-*l/ 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + 1\right) - 1 \]

      8. unpow2 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + 1\right) - 1 \]

   8. Simplified 62.5%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + 1\right)} - 1 \]
   9. Step-by-step derivation

      1. sub-neg 62.5%

         \[\leadsto \color{blue}{0.5 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} + 1\right) + \left(-1\right)} \]

      2. add-log-exp 62.4%

         \[\leadsto \color{blue}{\log \left(e^{0.5 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} + 1\right)}\right)} + \left(-1\right) \]

      3. *-commutative 62.4%

         \[\leadsto \log \left(e^{\color{blue}{\log \left({\left(\frac{x}{y}\right)}^{2} + 1\right) \cdot 0.5}}\right) + \left(-1\right) \]

      4. exp-to-pow 62.4%

         \[\leadsto \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{2} + 1\right)}^{0.5}\right)} + \left(-1\right) \]

      5. pow1/2 62.4%

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

      6. +-commutative 62.4%

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

      7. unpow2 62.4%

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

      8. hypot-1-def 63.3%

         \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right)} + \left(-1\right) \]

      9. metadata-eval 63.3%

         \[\leadsto \log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + \color{blue}{-1} \]

   10. Applied egg-rr 63.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)} \leq 2:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, \frac{x}{y}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \log \left(1 + \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 2.0) t_1 (+ -1.0 (* 0.5 (log (+ 1.0 (/ (/ x y) (/ y x)))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = -1.0 + (0.5 * log((1.0 + ((x / y) / (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) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if (t_1 <= 2.0d0) then
        tmp = t_1
    else
        tmp = (-1.0d0) + (0.5d0 * log((1.0d0 + ((x / y) / (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 t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = -1.0 + (0.5 * Math.log((1.0 + ((x / y) / (y / x)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_1
	else:
		tmp = -1.0 + (0.5 * math.log((1.0 + ((x / y) / (y / x)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(-1.0 + Float64(0.5 * log(Float64(1.0 + Float64(Float64(x / y) / Float64(y / x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = -1.0 + (0.5 * log((1.0 + ((x / y) / (y / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2.0], t$95$1, N[(-1.0 + N[(0.5 * N[Log[N[(1.0 + N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 99.6%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 47.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. pow2 47.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. add-log-exp 47.2%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{\frac{x \cdot x}{{y}^{2}}}\right)} - 1 \]

      3. add-sqr-sqrt 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{\sqrt{\frac{x \cdot x}{{y}^{2}}} \cdot \sqrt{\frac{x \cdot x}{{y}^{2}}}}}\right) - 1 \]

      4. pow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{\color{blue}{{\left(\sqrt{\frac{x \cdot x}{{y}^{2}}}\right)}^{2}}}\right) - 1 \]

      5. sqrt-div 47.2%

         \[\leadsto 0.5 \cdot \log \left(e^{{\color{blue}{\left(\frac{\sqrt{x \cdot x}}{\sqrt{{y}^{2}}}\right)}}^{2}}\right) - 1 \]

      6. sqrt-prod 22.3%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      7. add-sqr-sqrt 50.5%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{\color{blue}{x}}{\sqrt{{y}^{2}}}\right)}^{2}}\right) - 1 \]

      8. sqrt-pow1 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{{y}^{\left(\frac{2}{2}\right)}}}\right)}^{2}}\right) - 1 \]

      9. metadata-eval 60.0%

         \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{{y}^{\color{blue}{1}}}\right)}^{2}}\right) - 1 \]

      10. pow1 60.0%

          \[\leadsto 0.5 \cdot \log \left(e^{{\left(\frac{x}{\color{blue}{y}}\right)}^{2}}\right) - 1 \]

   5. Applied egg-rr 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\log \left(e^{{\left(\frac{x}{y}\right)}^{2}}\right)} - 1 \]

   6. Taylor expanded in x around 0 47.2%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(1 + \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   7. Step-by-step derivation

      1. +-commutative 47.2%

         \[\leadsto 0.5 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

      2. unpow2 47.2%

         \[\leadsto 0.5 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} + 1\right) - 1 \]

      3. associate-*r/ 51.4%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} + 1\right) - 1 \]

      4. unpow2 51.4%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \frac{x}{\color{blue}{y \cdot y}} + 1\right) - 1 \]

      5. associate-/l/ 62.0%

         \[\leadsto 0.5 \cdot \log \left(x \cdot \color{blue}{\frac{\frac{x}{y}}{y}} + 1\right) - 1 \]

      6. associate-*r/ 61.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x \cdot \frac{x}{y}}{y}} + 1\right) - 1 \]

      7. associate-*l/ 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + 1\right) - 1 \]

      8. unpow2 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + 1\right) - 1 \]

   8. Simplified 62.5%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + 1\right)} - 1 \]
   9. Step-by-step derivation

      1. unpow2 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + 1\right) - 1 \]

      2. clear-num 62.5%

         \[\leadsto 0.5 \cdot \log \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} + 1\right) - 1 \]

      3. un-div-inv 62.5%

         \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + 1\right) - 1 \]

   10. Applied egg-rr 62.5%

       \[\leadsto 0.5 \cdot \log \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + 1\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)} \leq 2:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \log \left(1 + \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-96}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-65} \lor \neg \left(x \leq 7.6 \cdot 10^{-22}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.6e-96)
   -1.0
   (if (or (<= x 1.9e-65) (not (<= x 7.6e-22)))
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.6e-96) {
		tmp = -1.0;
	} else if ((x <= 1.9e-65) || !(x <= 7.6e-22)) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} 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 <= 2.6d-96) then
        tmp = -1.0d0
    else if ((x <= 1.9d-65) .or. (.not. (x <= 7.6d-22))) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.6e-96) {
		tmp = -1.0;
	} else if ((x <= 1.9e-65) || !(x <= 7.6e-22)) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.6e-96:
		tmp = -1.0
	elif (x <= 1.9e-65) or not (x <= 7.6e-22):
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.6e-96)
		tmp = -1.0;
	elseif ((x <= 1.9e-65) || !(x <= 7.6e-22))
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.6e-96)
		tmp = -1.0;
	elseif ((x <= 1.9e-65) || ~((x <= 7.6e-22)))
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.6e-96], -1.0, If[Or[LessEqual[x, 1.9e-65], N[Not[LessEqual[x, 7.6e-22]], $MachinePrecision]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000002e-96 or 1.9000000000000001e-65 < x < 7.60000000000000046e-22

   1. Initial program 53.6%

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

   3. Taylor expanded in x around 0 64.2%

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

   if 2.6000000000000002e-96 < x < 1.9000000000000001e-65 or 7.60000000000000046e-22 < x

   1. Initial program 61.0%

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

   3. Taylor expanded in y around 0 67.1%

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

      1. unpow2 67.1%

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

      2. pow2 67.1%

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

      3. times-frac 72.6%

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

   5. Applied egg-rr 72.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-96}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-65} \lor \neg \left(x \leq 7.6 \cdot 10^{-22}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-73} \lor \neg \left(y \leq 2.8 \cdot 10^{-19}\right) \land y \leq 6400000000:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 2.8e-73) (and (not (<= y 2.8e-19)) (<= y 6400000000.0)))
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
   (+ -1.0 (* 0.5 (* (/ x y) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 2.8e-73) || (!(y <= 2.8e-19) && (y <= 6400000000.0))) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 2.8d-73) .or. (.not. (y <= 2.8d-19)) .and. (y <= 6400000000.0d0)) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else
        tmp = (-1.0d0) + (0.5d0 * ((x / y) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 2.8e-73) || (!(y <= 2.8e-19) && (y <= 6400000000.0))) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 2.8e-73) or (not (y <= 2.8e-19) and (y <= 6400000000.0)):
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 2.8e-73) || (!(y <= 2.8e-19) && (y <= 6400000000.0)))
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 2.8e-73) || (~((y <= 2.8e-19)) && (y <= 6400000000.0)))
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 2.8e-73], And[N[Not[LessEqual[y, 2.8e-19]], $MachinePrecision], LessEqual[y, 6400000000.0]]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.80000000000000012e-73 or 2.80000000000000003e-19 < y < 6.4e9

   1. Initial program 60.4%

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

   3. Taylor expanded in y around 0 47.6%

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

      1. unpow2 47.6%

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

      2. pow2 47.6%

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

      3. times-frac 54.4%

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

   5. Applied egg-rr 54.4%

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

   if 2.80000000000000012e-73 < y < 2.80000000000000003e-19 or 6.4e9 < y

   1. Initial program 43.6%

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

   3. Taylor expanded in x around 0 74.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. pow2 74.0%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 74.0%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 76.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

   5. Applied egg-rr 76.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-73} \lor \neg \left(y \leq 2.8 \cdot 10^{-19}\right) \land y \leq 6400000000:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-95}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3e-95) -1.0 (if (<= x 1.85e-65) 1.0 (if (<= x 5e-30) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3e-95) {
		tmp = -1.0;
	} else if (x <= 1.85e-65) {
		tmp = 1.0;
	} else if (x <= 5e-30) {
		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 <= 3d-95) then
        tmp = -1.0d0
    else if (x <= 1.85d-65) then
        tmp = 1.0d0
    else if (x <= 5d-30) 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 <= 3e-95) {
		tmp = -1.0;
	} else if (x <= 1.85e-65) {
		tmp = 1.0;
	} else if (x <= 5e-30) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3e-95:
		tmp = -1.0
	elif x <= 1.85e-65:
		tmp = 1.0
	elif x <= 5e-30:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3e-95)
		tmp = -1.0;
	elseif (x <= 1.85e-65)
		tmp = 1.0;
	elseif (x <= 5e-30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3e-95)
		tmp = -1.0;
	elseif (x <= 1.85e-65)
		tmp = 1.0;
	elseif (x <= 5e-30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3e-95], -1.0, If[LessEqual[x, 1.85e-65], 1.0, If[LessEqual[x, 5e-30], -1.0, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e-95 or 1.85e-65 < x < 4.99999999999999972e-30

   1. Initial program 53.6%

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

   3. Taylor expanded in x around 0 64.2%

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

   if 3e-95 < x < 1.85e-65 or 4.99999999999999972e-30 < x

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf 69.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-95}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% 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 55.3%

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

3. Taylor expanded in x around 0 55.8%

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

4. Final simplification 55.8%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 51.4% 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 2024078 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (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))))