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

Average Error: 31.9 → 13.1

Time: 3.8s

Precision: binary64

Cost: 1744

Math

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

↓

\[\begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+98}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+69}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y)) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= y -1.1e+98)
     -1.0
     (if (<= y -5.9e-125)
       t_1
       (if (<= y 1e-156)
         1.0
         (if (<= y 7.5e+18) t_1 (if (<= y 1.15e+69) 1.0 -1.0)))))))

C

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

↓

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (y <= -1.1e+98) {
		tmp = -1.0;
	} else if (y <= -5.9e-125) {
		tmp = t_1;
	} else if (y <= 1e-156) {
		tmp = 1.0;
	} else if (y <= 7.5e+18) {
		tmp = t_1;
	} else if (y <= 1.15e+69) {
		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
    code = ((x * x) - ((y * 4.0d0) * y)) / ((x * x) + ((y * 4.0d0) * y))
end function

↓

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 * 4.0d0) * y
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if (y <= (-1.1d+98)) then
        tmp = -1.0d0
    else if (y <= (-5.9d-125)) then
        tmp = t_1
    else if (y <= 1d-156) then
        tmp = 1.0d0
    else if (y <= 7.5d+18) then
        tmp = t_1
    else if (y <= 1.15d+69) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (y <= -1.1e+98) {
		tmp = -1.0;
	} else if (y <= -5.9e-125) {
		tmp = t_1;
	} else if (y <= 1e-156) {
		tmp = 1.0;
	} else if (y <= 7.5e+18) {
		tmp = t_1;
	} else if (y <= 1.15e+69) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (y * 4.0) * y
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if y <= -1.1e+98:
		tmp = -1.0
	elif y <= -5.9e-125:
		tmp = t_1
	elif y <= 1e-156:
		tmp = 1.0
	elif y <= 7.5e+18:
		tmp = t_1
	elif y <= 1.15e+69:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (y <= -1.1e+98)
		tmp = -1.0;
	elseif (y <= -5.9e-125)
		tmp = t_1;
	elseif (y <= 1e-156)
		tmp = 1.0;
	elseif (y <= 7.5e+18)
		tmp = t_1;
	elseif (y <= 1.15e+69)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = (y * 4.0) * y;
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if (y <= -1.1e+98)
		tmp = -1.0;
	elseif (y <= -5.9e-125)
		tmp = t_1;
	elseif (y <= 1e-156)
		tmp = 1.0;
	elseif (y <= 7.5e+18)
		tmp = t_1;
	elseif (y <= 1.15e+69)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $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[y, -1.1e+98], -1.0, If[LessEqual[y, -5.9e-125], t$95$1, If[LessEqual[y, 1e-156], 1.0, If[LessEqual[y, 7.5e+18], t$95$1, If[LessEqual[y, 1.15e+69], 1.0, -1.0]]]]]]]

Target

Original	31.9
Target	31.7
Herbie	13.1

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

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000004e98 or 1.15000000000000008e69 < y

   1. Initial program 49.4

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

   2. Taylor expanded in x around 0 11.6

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

   if -1.10000000000000004e98 < y < -5.89999999999999959e-125 or 1.00000000000000004e-156 < y < 7.5e18

   1. Initial program 15.5

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

   if -5.89999999999999959e-125 < y < 1.00000000000000004e-156 or 7.5e18 < y < 1.15000000000000008e69

   1. Initial program 27.8

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

   2. Taylor expanded in x around inf 12.3

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+98}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;y \leq 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+69}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	16.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	31.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023090 
(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))))