Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A

Average Error: 23.38% → 0.22%

Time: 9.9s

Precision: binary64

Cost: 6784

Math

\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

↓

\[\frac{-\tan \left(\frac{x}{2}\right)}{-0.75} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))

↓

(FPCore (x) :precision binary64 (/ (- (tan (/ x 2.0))) -0.75))

C

double code(double x) {
	return (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
}

↓

double code(double x) {
	return -tan((x / 2.0)) / -0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((8.0d0 / 3.0d0) * sin((x * 0.5d0))) * sin((x * 0.5d0))) / sin(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = -tan((x / 2.0d0)) / (-0.75d0)
end function

Java

public static double code(double x) {
	return (((8.0 / 3.0) * Math.sin((x * 0.5))) * Math.sin((x * 0.5))) / Math.sin(x);
}

↓

public static double code(double x) {
	return -Math.tan((x / 2.0)) / -0.75;
}

Python

def code(x):
	return (((8.0 / 3.0) * math.sin((x * 0.5))) * math.sin((x * 0.5))) / math.sin(x)

↓

def code(x):
	return -math.tan((x / 2.0)) / -0.75

Julia

function code(x)
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * sin(Float64(x * 0.5))) * sin(Float64(x * 0.5))) / sin(x))
end

↓

function code(x)
	return Float64(Float64(-tan(Float64(x / 2.0))) / -0.75)
end

MATLAB

function tmp = code(x)
	tmp = (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
end

↓

function tmp = code(x)
	tmp = -tan((x / 2.0)) / -0.75;
end

Wolfram

code[x_] := N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[((-N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]) / -0.75), $MachinePrecision]

Target

Original	23.38%
Target	0.48%
Herbie	0.22%

\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]

Derivation

1. Initial program 23.38

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

2. Simplified 23.38

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot 2.6666666666666665} \]
   Proof

   [Start] 23.38	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*l* [=>] 23.42	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
   associate-*r/ [<=] 23.38	\[ \color{blue}{\frac{8}{3} \cdot \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]
   *-commutative [<=] 23.38	\[ \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8}{3}} \]
   metadata-eval [=>] 23.38	\[ \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \color{blue}{2.6666666666666665} \]

3. Applied egg-rr 61.65

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}\right)} - 1} \]

4. Simplified 46.97

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x \cdot 0.75}} \]
   Proof

   [Start] 61.65	\[ e^{\mathsf{log1p}\left(\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}\right)} - 1 \]
   expm1-def [=>] 61.63	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}\right)\right)} \]
   expm1-log1p [=>] 46.96	\[ \color{blue}{\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}} \]
   associate-/l/ [=>] 46.97	\[ \color{blue}{\frac{\cos 0 - \cos x}{\sin x \cdot 0.75}} \]
   cos-0 [=>] 46.97	\[ \frac{\color{blue}{1} - \cos x}{\sin x \cdot 0.75} \]

5. Applied egg-rr 0.22

   \[\leadsto \color{blue}{-\frac{\tan \left(\frac{x}{2}\right)}{-0.75}} \]

6. Simplified 0.22

   \[\leadsto \color{blue}{\frac{-\tan \left(\frac{x}{2}\right)}{-0.75}} \]
   Proof

   [Start] 0.22	\[ -\frac{\tan \left(\frac{x}{2}\right)}{-0.75} \]
   distribute-neg-frac [=>] 0.22	\[ \color{blue}{\frac{-\tan \left(\frac{x}{2}\right)}{-0.75}} \]

7. Final simplification 0.22

   \[\leadsto \frac{-\tan \left(\frac{x}{2}\right)}{-0.75} \]

Alternatives

Alternative 1
Error	42.59%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\frac{8}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \end{array} \]

Alternative 2
Error	0.57%
Cost	6720

\[\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333 \]

Alternative 3
Error	49.33%
Cost	320

\[\frac{0.2962962962962963}{\frac{0.4444444444444444}{x}} \]

Alternative 4
Error	49.14%
Cost	320

\[\frac{x \cdot 0.25}{0.375} \]

Alternative 5
Error	49.4%
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))