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

Percentage Accurate: 50.6% → 74.9%

Time: 7.1s

Alternatives: 5

Speedup: 2.6×

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: 50.6% 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: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 62000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 3.3 \cdot 10^{+62} \lor \neg \left(y_m \leq 4.6 \cdot 10^{+86}\right):\\ \;\;\;\;0.5 \cdot \frac{x \cdot \frac{x}{y_m}}{y_m} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 62000000000.0)
   1.0
   (if (or (<= y_m 3.3e+62) (not (<= y_m 4.6e+86)))
     (+ (* 0.5 (/ (* x (/ x y_m)) y_m)) -1.0)
     1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 62000000000.0) {
		tmp = 1.0;
	} else if ((y_m <= 3.3e+62) || !(y_m <= 4.6e+86)) {
		tmp = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 62000000000.0d0) then
        tmp = 1.0d0
    else if ((y_m <= 3.3d+62) .or. (.not. (y_m <= 4.6d+86))) then
        tmp = (0.5d0 * ((x * (x / y_m)) / y_m)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 62000000000.0) {
		tmp = 1.0;
	} else if ((y_m <= 3.3e+62) || !(y_m <= 4.6e+86)) {
		tmp = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 62000000000.0:
		tmp = 1.0
	elif (y_m <= 3.3e+62) or not (y_m <= 4.6e+86):
		tmp = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 62000000000.0)
		tmp = 1.0;
	elseif ((y_m <= 3.3e+62) || !(y_m <= 4.6e+86))
		tmp = Float64(Float64(0.5 * Float64(Float64(x * Float64(x / y_m)) / y_m)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 62000000000.0)
		tmp = 1.0;
	elseif ((y_m <= 3.3e+62) || ~((y_m <= 4.6e+86)))
		tmp = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 62000000000.0], 1.0, If[Or[LessEqual[y$95$m, 3.3e+62], N[Not[LessEqual[y$95$m, 4.6e+86]], $MachinePrecision]], N[(N[(0.5 * N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2e10 or 3.3e62 < y < 4.59999999999999979e86

   1. Initial program 56.9%

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

   2. Taylor expanded in x around inf 59.1%

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

   if 6.2e10 < y < 3.3e62 or 4.59999999999999979e86 < y

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) - 1} \]

   3. Taylor expanded in x around 0 79.7%

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

      1. unpow2 79.7%

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

      2. unpow2 79.7%

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

      3. times-frac 91.1%

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

      4. unpow2 91.1%

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

   5. Simplified 91.1%

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

      1. pow2 91.1%

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

      2. associate-*r/ 91.0%

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

   7. Applied egg-rr 91.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 62000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+62} \lor \neg \left(y \leq 4.6 \cdot 10^{+86}\right):\\ \;\;\;\;0.5 \cdot \frac{x \cdot \frac{x}{y}}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 76.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* y_m (* y_m 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 2.0) t_1 (+ (* 0.5 (pow (/ x y_m) 2.0)) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = y_m * (y_m * 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 = (0.5 * pow((x / y_m), 2.0)) + -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y_m * (y_m * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if (t_1 <= 2.0d0) then
        tmp = t_1
    else
        tmp = (0.5d0 * ((x / y_m) ** 2.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = y_m * (y_m * 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 = (0.5 * Math.pow((x / y_m), 2.0)) + -1.0;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(y_m * Float64(y_m * 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(Float64(0.5 * (Float64(x / y_m) ^ 2.0)) + -1.0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = y_m * (y_m * 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 = (0.5 * ((x / y_m) ^ 2.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(y$95$m * 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[(0.5 * N[Power[N[(x / y$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) 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. Taylor expanded in x around 0 34.1%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) - 1} \]

   3. Taylor expanded in x around 0 40.8%

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

      1. unpow2 40.8%

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

      2. unpow2 40.8%

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

      3. times-frac 60.1%

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

      4. unpow2 60.1%

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

   5. Simplified 60.1%

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

4. Final simplification 80.8%

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

Alternative 3: 76.8% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* y_m (* y_m 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 2.0) t_1 (+ (* 0.5 (/ (* x (/ x y_m)) y_m)) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = y_m * (y_m * 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 = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y_m * (y_m * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if (t_1 <= 2.0d0) then
        tmp = t_1
    else
        tmp = (0.5d0 * ((x * (x / y_m)) / y_m)) + (-1.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = y_m * (y_m * 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 = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(y_m * Float64(y_m * 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(Float64(0.5 * Float64(Float64(x * Float64(x / y_m)) / y_m)) + -1.0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = y_m * (y_m * 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 = (0.5 * ((x * (x / y_m)) / y_m)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(y$95$m * 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[(0.5 * N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) 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. Taylor expanded in x around 0 34.1%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) - 1} \]

   3. Taylor expanded in x around 0 40.8%

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

      1. unpow2 40.8%

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

      2. unpow2 40.8%

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

      3. times-frac 60.1%

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

      4. unpow2 60.1%

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

   5. Simplified 60.1%

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

      1. pow2 60.1%

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

      2. associate-*r/ 60.0%

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

   7. Applied egg-rr 60.0%

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

4. Final simplification 80.8%

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

Alternative 4: 74.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 44000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 2 \cdot 10^{+62}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y_m \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 44000000000.0)
   1.0
   (if (<= y_m 2e+62) -1.0 (if (<= y_m 4.4e+86) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 44000000000.0) {
		tmp = 1.0;
	} else if (y_m <= 2e+62) {
		tmp = -1.0;
	} else if (y_m <= 4.4e+86) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 44000000000.0d0) then
        tmp = 1.0d0
    else if (y_m <= 2d+62) then
        tmp = -1.0d0
    else if (y_m <= 4.4d+86) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 44000000000.0) {
		tmp = 1.0;
	} else if (y_m <= 2e+62) {
		tmp = -1.0;
	} else if (y_m <= 4.4e+86) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 44000000000.0:
		tmp = 1.0
	elif y_m <= 2e+62:
		tmp = -1.0
	elif y_m <= 4.4e+86:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 44000000000.0)
		tmp = 1.0;
	elseif (y_m <= 2e+62)
		tmp = -1.0;
	elseif (y_m <= 4.4e+86)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 44000000000.0)
		tmp = 1.0;
	elseif (y_m <= 2e+62)
		tmp = -1.0;
	elseif (y_m <= 4.4e+86)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 44000000000.0], 1.0, If[LessEqual[y$95$m, 2e+62], -1.0, If[LessEqual[y$95$m, 4.4e+86], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4e10 or 2.00000000000000007e62 < y < 4.40000000000000006e86

   1. Initial program 56.9%

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

   2. Taylor expanded in x around inf 59.1%

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

   if 4.4e10 < y < 2.00000000000000007e62 or 4.40000000000000006e86 < y

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 89.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 44000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+62}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 50.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

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

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 51.9%

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

2. Taylor expanded in x around 0 51.7%

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

3. Final simplification 51.7%

   \[\leadsto -1 \]

Developer target: 51.0% 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 2023320 
(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))))