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

Average Error: 49.9% → 20.04%

Time: 5.9s

Precision: binary64

Cost: 1744

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ t_1 := -1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+69}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (+ (* x x) (* y (* y -4.0))) (+ (* x x) (* y (* y 4.0)))))
        (t_1 (+ -1.0 (/ 0.5 (/ (/ y x) (/ x y))))))
   (if (<= y -1.1e+98)
     t_1
     (if (<= y -5.9e-125)
       t_0
       (if (<= y 1e-156)
         1.0
         (if (<= y 7.5e+18) t_0 (if (<= y 1.15e+69) 1.0 t_1)))))))

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 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	double t_1 = -1.0 + (0.5 / ((y / x) / (x / y)));
	double tmp;
	if (y <= -1.1e+98) {
		tmp = t_1;
	} else if (y <= -5.9e-125) {
		tmp = t_0;
	} else if (y <= 1e-156) {
		tmp = 1.0;
	} else if (y <= 7.5e+18) {
		tmp = t_0;
	} else if (y <= 1.15e+69) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	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 = ((x * x) + (y * (y * (-4.0d0)))) / ((x * x) + (y * (y * 4.0d0)))
    t_1 = (-1.0d0) + (0.5d0 / ((y / x) / (x / y)))
    if (y <= (-1.1d+98)) then
        tmp = t_1
    else if (y <= (-5.9d-125)) then
        tmp = t_0
    else if (y <= 1d-156) then
        tmp = 1.0d0
    else if (y <= 7.5d+18) then
        tmp = t_0
    else if (y <= 1.15d+69) then
        tmp = 1.0d0
    else
        tmp = t_1
    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 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	double t_1 = -1.0 + (0.5 / ((y / x) / (x / y)));
	double tmp;
	if (y <= -1.1e+98) {
		tmp = t_1;
	} else if (y <= -5.9e-125) {
		tmp = t_0;
	} else if (y <= 1e-156) {
		tmp = 1.0;
	} else if (y <= 7.5e+18) {
		tmp = t_0;
	} else if (y <= 1.15e+69) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	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 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)))
	t_1 = -1.0 + (0.5 / ((y / x) / (x / y)))
	tmp = 0
	if y <= -1.1e+98:
		tmp = t_1
	elif y <= -5.9e-125:
		tmp = t_0
	elif y <= 1e-156:
		tmp = 1.0
	elif y <= 7.5e+18:
		tmp = t_0
	elif y <= 1.15e+69:
		tmp = 1.0
	else:
		tmp = t_1
	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(Float64(x * x) + Float64(y * Float64(y * -4.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))))
	t_1 = Float64(-1.0 + Float64(0.5 / Float64(Float64(y / x) / Float64(x / y))))
	tmp = 0.0
	if (y <= -1.1e+98)
		tmp = t_1;
	elseif (y <= -5.9e-125)
		tmp = t_0;
	elseif (y <= 1e-156)
		tmp = 1.0;
	elseif (y <= 7.5e+18)
		tmp = t_0;
	elseif (y <= 1.15e+69)
		tmp = 1.0;
	else
		tmp = t_1;
	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 = ((x * x) + (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	t_1 = -1.0 + (0.5 / ((y / x) / (x / y)));
	tmp = 0.0;
	if (y <= -1.1e+98)
		tmp = t_1;
	elseif (y <= -5.9e-125)
		tmp = t_0;
	elseif (y <= 1e-156)
		tmp = 1.0;
	elseif (y <= 7.5e+18)
		tmp = t_0;
	elseif (y <= 1.15e+69)
		tmp = 1.0;
	else
		tmp = t_1;
	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[(N[(x * x), $MachinePrecision] + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+98], t$95$1, If[LessEqual[y, -5.9e-125], t$95$0, If[LessEqual[y, 1e-156], 1.0, If[LessEqual[y, 7.5e+18], t$95$0, If[LessEqual[y, 1.15e+69], 1.0, t$95$1]]]]]]]

Target

Original	49.9%
Target	49.49%
Herbie	20.04%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 77.23

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

   2. Taylor expanded in x around 0 36.04

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

   3. Simplified 36.04

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

      [Start] 36.04	\[ \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) - 1 \]
      sub-neg [=>] 36.04	\[ \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) + \left(-1\right)} \]
      metadata-eval [=>] 36.04	\[ \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) + \color{blue}{-1} \]
      +-commutative [=>] 36.04	\[ \color{blue}{-1 + \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right)} \]
      fma-def [=>] 36.04	\[ -1 + \color{blue}{\mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right)} \]
      unpow2 [=>] 36.04	\[ -1 + \mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}}\right) \]
      associate-*r/ [=>] 36.04	\[ -1 + \mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, \color{blue}{\frac{0.5 \cdot \left(x \cdot x\right)}{{y}^{2}}}\right) \]
      associate-/l* [=>] 36.04	\[ -1 + \mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, \color{blue}{\frac{0.5}{\frac{{y}^{2}}{x \cdot x}}}\right) \]
      unpow2 [=>] 36.04	\[ -1 + \mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, \frac{0.5}{\frac{\color{blue}{y \cdot y}}{x \cdot x}}\right) \]
      associate-/l* [=>] 36.04	\[ -1 + \mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, \frac{0.5}{\color{blue}{\frac{y}{\frac{x \cdot x}{y}}}}\right) \]

   4. Taylor expanded in x around 0 26.66

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

   5. Simplified 17.18

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

      [Start] 26.66	\[ -1 + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} \]
      associate-*r/ [=>] 26.66	\[ -1 + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{y}^{2}}} \]
      unpow2 [=>] 26.66	\[ -1 + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{{y}^{2}} \]
      unpow2 [=>] 26.66	\[ -1 + \frac{0.5 \cdot \left(x \cdot x\right)}{\color{blue}{y \cdot y}} \]
      associate-/l* [=>] 26.66	\[ -1 + \color{blue}{\frac{0.5}{\frac{y \cdot y}{x \cdot x}}} \]
      times-frac [=>] 17.18	\[ -1 + \frac{0.5}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]
      unpow2 [<=] 17.18	\[ -1 + \frac{0.5}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   6. Applied egg-rr 17.18

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

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

   1. Initial program 24.23

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

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

   1. Initial program 43.45

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

   2. Taylor expanded in x around inf 19.17

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

4. Final simplification 20.04

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+98}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;y \leq -5.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;y \leq 10^{-156}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+69}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.9%
Cost	1869

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+38} \lor \neg \left(t_0 \leq 10^{+107}\right):\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	25.44%
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 3
Error	49.67%
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))))