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

Percentage Accurate: 51.2% → 79.3%

Time: 6.7s

Alternatives: 5

Speedup: 1.7×

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: 51.2% 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: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 3 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (fma 0.5 (* (/ x y) (/ x y)) -1.0)))
   (if (<= (* x x) 5e-173)
     t_1
     (if (<= (* x x) 3e+188)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 2e+238) t_1 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma(0.5, ((x / y) * (x / y)), -1.0);
	double tmp;
	if ((x * x) <= 5e-173) {
		tmp = t_1;
	} else if ((x * x) <= 3e+188) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+238) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0)
	tmp = 0.0
	if (Float64(x * x) <= 5e-173)
		tmp = t_1;
	elseif (Float64(x * x) <= 3e+188)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 2e+238)
		tmp = t_1;
	else
		tmp = 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[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-173], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 3e+188], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+238], t$95$1, 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000002e-173 or 3.0000000000000001e188 < (*.f64 x x) < 2.0000000000000001e238

   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. Taylor expanded in x around 0 80.9%

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

      1. fma-neg 80.9%

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

      2. unpow2 80.9%

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

      3. unpow2 80.9%

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

      4. times-frac 87.3%

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

      5. metadata-eval 87.3%

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

   4. Simplified 87.3%

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

   if 5.0000000000000002e-173 < (*.f64 x x) < 3.0000000000000001e188

   1. Initial program 76.8%

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

   if 2.0000000000000001e238 < (*.f64 x x)

   1. Initial program 18.9%

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

   2. Taylor expanded in x around inf 93.4%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 3 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\frac{\left(y \cdot y\right) \cdot -8}{x}}{x}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+57}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+238}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (/ (* (* y y) -8.0) x) x))))
   (if (<= (* x x) 5e-173)
     -1.0
     (if (<= (* x x) 4e-71)
       t_0
       (if (<= (* x x) 2e+57)
         -1.0
         (if (<= (* x x) 3e+188) t_0 (if (<= (* x x) 2e+238) -1.0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((((y * y) * -8.0) / x) / x);
	double tmp;
	if ((x * x) <= 5e-173) {
		tmp = -1.0;
	} else if ((x * x) <= 4e-71) {
		tmp = t_0;
	} else if ((x * x) <= 2e+57) {
		tmp = -1.0;
	} else if ((x * x) <= 3e+188) {
		tmp = t_0;
	} else if ((x * x) <= 2e+238) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((((y * y) * (-8.0d0)) / x) / x)
    if ((x * x) <= 5d-173) then
        tmp = -1.0d0
    else if ((x * x) <= 4d-71) then
        tmp = t_0
    else if ((x * x) <= 2d+57) then
        tmp = -1.0d0
    else if ((x * x) <= 3d+188) then
        tmp = t_0
    else if ((x * x) <= 2d+238) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((((y * y) * -8.0) / x) / x);
	double tmp;
	if ((x * x) <= 5e-173) {
		tmp = -1.0;
	} else if ((x * x) <= 4e-71) {
		tmp = t_0;
	} else if ((x * x) <= 2e+57) {
		tmp = -1.0;
	} else if ((x * x) <= 3e+188) {
		tmp = t_0;
	} else if ((x * x) <= 2e+238) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((((y * y) * -8.0) / x) / x)
	tmp = 0
	if (x * x) <= 5e-173:
		tmp = -1.0
	elif (x * x) <= 4e-71:
		tmp = t_0
	elif (x * x) <= 2e+57:
		tmp = -1.0
	elif (x * x) <= 3e+188:
		tmp = t_0
	elif (x * x) <= 2e+238:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(Float64(Float64(y * y) * -8.0) / x) / x))
	tmp = 0.0
	if (Float64(x * x) <= 5e-173)
		tmp = -1.0;
	elseif (Float64(x * x) <= 4e-71)
		tmp = t_0;
	elseif (Float64(x * x) <= 2e+57)
		tmp = -1.0;
	elseif (Float64(x * x) <= 3e+188)
		tmp = t_0;
	elseif (Float64(x * x) <= 2e+238)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((((y * y) * -8.0) / x) / x);
	tmp = 0.0;
	if ((x * x) <= 5e-173)
		tmp = -1.0;
	elseif ((x * x) <= 4e-71)
		tmp = t_0;
	elseif ((x * x) <= 2e+57)
		tmp = -1.0;
	elseif ((x * x) <= 3e+188)
		tmp = t_0;
	elseif ((x * x) <= 2e+238)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(N[(N[(y * y), $MachinePrecision] * -8.0), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-173], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 4e-71], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 2e+57], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 3e+188], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 2e+238], -1.0, 1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000002e-173 or 3.9999999999999997e-71 < (*.f64 x x) < 2.0000000000000001e57 or 3.0000000000000001e188 < (*.f64 x x) < 2.0000000000000001e238

   1. Initial program 57.9%

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

   2. Taylor expanded in x around 0 85.3%

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

   if 5.0000000000000002e-173 < (*.f64 x x) < 3.9999999999999997e-71 or 2.0000000000000001e57 < (*.f64 x x) < 3.0000000000000001e188

   1. Initial program 80.8%

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

   2. Taylor expanded in x around inf 70.5%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 70.5%

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

      2. unpow2 70.5%

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

      3. associate-*r/ 70.5%

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

      4. *-commutative 70.5%

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

      5. unpow2 70.5%

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

      6. associate-*r* 70.5%

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

      7. unpow2 70.5%

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

      8. associate-*r/ 70.5%

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

      9. *-commutative 70.5%

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

      10. unpow2 70.5%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{\left(y \cdot y\right)} \cdot 4}{x \cdot x}\right) \]

      11. associate-*r* 70.5%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x}\right) \]

   4. Simplified 70.5%

      \[\leadsto \color{blue}{1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right)} \]
   5. Step-by-step derivation

      1. sub-div 70.5%

         \[\leadsto 1 + \color{blue}{\frac{y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)}{x \cdot x}} \]

      2. associate-/r* 70.5%

         \[\leadsto 1 + \color{blue}{\frac{\frac{y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)}{x}}{x}} \]

      3. associate-*r* 70.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{\left(y \cdot y\right) \cdot -4} - y \cdot \left(y \cdot 4\right)}{x}}{x} \]

      4. associate-*r* 70.5%

         \[\leadsto 1 + \frac{\frac{\left(y \cdot y\right) \cdot -4 - \color{blue}{\left(y \cdot y\right) \cdot 4}}{x}}{x} \]

      5. distribute-lft-out-- 70.5%

         \[\leadsto 1 + \frac{\frac{\color{blue}{\left(y \cdot y\right) \cdot \left(-4 - 4\right)}}{x}}{x} \]

      6. metadata-eval 70.5%

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

   6. Applied egg-rr 70.5%

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

   if 2.0000000000000001e238 < (*.f64 x x)

   1. Initial program 18.9%

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

   2. Taylor expanded in x around inf 93.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{-71}:\\ \;\;\;\;1 + \frac{\frac{\left(y \cdot y\right) \cdot -8}{x}}{x}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+57}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3 \cdot 10^{+188}:\\ \;\;\;\;1 + \frac{\frac{\left(y \cdot y\right) \cdot -8}{x}}{x}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+238}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+238}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-173)
     -1.0
     (if (<= (* x x) 3e+188)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 2e+238) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-173) {
		tmp = -1.0;
	} else if ((x * x) <= 3e+188) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+238) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 5d-173) then
        tmp = -1.0d0
    else if ((x * x) <= 3d+188) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else if ((x * x) <= 2d+238) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    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) <= 5e-173) {
		tmp = -1.0;
	} else if ((x * x) <= 3e+188) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+238) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 5e-173:
		tmp = -1.0
	elif (x * x) <= 3e+188:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	elif (x * x) <= 2e+238:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-173)
		tmp = -1.0;
	elseif (Float64(x * x) <= 3e+188)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 2e+238)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 5e-173)
		tmp = -1.0;
	elseif ((x * x) <= 3e+188)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	elseif ((x * x) <= 2e+238)
		tmp = -1.0;
	else
		tmp = 1.0;
	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], 5e-173], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 3e+188], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+238], -1.0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.0000000000000002e-173 or 3.0000000000000001e188 < (*.f64 x x) < 2.0000000000000001e238

   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. Taylor expanded in x around 0 86.8%

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

   if 5.0000000000000002e-173 < (*.f64 x x) < 3.0000000000000001e188

   1. Initial program 76.8%

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

   if 2.0000000000000001e238 < (*.f64 x x)

   1. Initial program 18.9%

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

   2. Taylor expanded in x around inf 93.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+238}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 61.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+119}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.55e-80)
   -1.0
   (if (<= x 4.8e-39)
     1.0
     (if (<= x 1e+46)
       -1.0
       (if (<= x 1.8e+94) 1.0 (if (<= x 3.4e+119) -1.0 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.55e-80) {
		tmp = -1.0;
	} else if (x <= 4.8e-39) {
		tmp = 1.0;
	} else if (x <= 1e+46) {
		tmp = -1.0;
	} else if (x <= 1.8e+94) {
		tmp = 1.0;
	} else if (x <= 3.4e+119) {
		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 <= 1.55d-80) then
        tmp = -1.0d0
    else if (x <= 4.8d-39) then
        tmp = 1.0d0
    else if (x <= 1d+46) then
        tmp = -1.0d0
    else if (x <= 1.8d+94) then
        tmp = 1.0d0
    else if (x <= 3.4d+119) 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 <= 1.55e-80) {
		tmp = -1.0;
	} else if (x <= 4.8e-39) {
		tmp = 1.0;
	} else if (x <= 1e+46) {
		tmp = -1.0;
	} else if (x <= 1.8e+94) {
		tmp = 1.0;
	} else if (x <= 3.4e+119) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.55e-80:
		tmp = -1.0
	elif x <= 4.8e-39:
		tmp = 1.0
	elif x <= 1e+46:
		tmp = -1.0
	elif x <= 1.8e+94:
		tmp = 1.0
	elif x <= 3.4e+119:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.55e-80)
		tmp = -1.0;
	elseif (x <= 4.8e-39)
		tmp = 1.0;
	elseif (x <= 1e+46)
		tmp = -1.0;
	elseif (x <= 1.8e+94)
		tmp = 1.0;
	elseif (x <= 3.4e+119)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.55e-80)
		tmp = -1.0;
	elseif (x <= 4.8e-39)
		tmp = 1.0;
	elseif (x <= 1e+46)
		tmp = -1.0;
	elseif (x <= 1.8e+94)
		tmp = 1.0;
	elseif (x <= 3.4e+119)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.55e-80], -1.0, If[LessEqual[x, 4.8e-39], 1.0, If[LessEqual[x, 1e+46], -1.0, If[LessEqual[x, 1.8e+94], 1.0, If[LessEqual[x, 3.4e+119], -1.0, 1.0]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000008e-80 or 4.80000000000000031e-39 < x < 9.9999999999999999e45 or 1.79999999999999996e94 < x < 3.40000000000000013e119

   1. Initial program 50.2%

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

   2. Taylor expanded in x around 0 58.4%

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

   if 1.55000000000000008e-80 < x < 4.80000000000000031e-39 or 9.9999999999999999e45 < x < 1.79999999999999996e94 or 3.40000000000000013e119 < x

   1. Initial program 43.9%

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

   2. Taylor expanded in x around inf 93.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+119}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 50.8% 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 48.8%

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

2. Taylor expanded in x around 0 47.3%

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

3. Final simplification 47.3%

   \[\leadsto -1 \]

Developer target: 51.7% 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 2023283 
(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))))