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

Percentage Accurate: 51.5% → 81.0%

Time: 8.5s

Alternatives: 3

Speedup: 0.9×

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: 51.5% 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: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;x\_m \leq 5.6 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, x\_m \cdot \frac{x\_m}{y}, -1\right)\\ \mathbf{elif}\;x\_m \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y)))
   (if (<= x_m 5.6e-130)
     (fma (/ 0.5 y) (* x_m (/ x_m y)) -1.0)
     (if (<= x_m 2e+111) (/ (- (* x_m x_m) t_0) (+ (* x_m x_m) t_0)) 1.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (y * 4.0) * y;
	double tmp;
	if (x_m <= 5.6e-130) {
		tmp = fma((0.5 / y), (x_m * (x_m / y)), -1.0);
	} else if (x_m <= 2e+111) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (x_m <= 5.6e-130)
		tmp = fma(Float64(0.5 / y), Float64(x_m * Float64(x_m / y)), -1.0);
	elseif (x_m <= 2e+111)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(Float64(x_m * x_m) + t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x$95$m, 5.6e-130], N[(N[(0.5 / y), $MachinePrecision] * N[(x$95$m * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[x$95$m, 2e+111], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.60000000000000032e-130

   1. Initial program 54.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. *-inverses N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if 5.60000000000000032e-130 < x < 1.99999999999999991e111

   1. Initial program 78.1%

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

   if 1.99999999999999991e111 < x

   1. Initial program 14.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 89.5%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0} \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (y * 4.0) * y;
	double tmp;
	if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= -1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], -1.0, 1.0]]

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))) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   if -1 < (/.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 39.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.6%

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

Alternative 3: 49.6% accurate, 48.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := -1.0

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 50.1%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 51.9% accurate, 0.2× 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 2024339 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

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