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

Percentage Accurate: 76.0% → 99.8%

Time: 10.9s

Precision: binary64

Cost: 6720

Math

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

↓

\[\frac{\tan \left(x \cdot 0.5\right)}{0.75} \]

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 0.5)) 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 * 0.5)) / 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 * 0.5d0)) / 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 * 0.5)) / 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 * 0.5)) / 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(tan(Float64(x * 0.5)) / 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 * 0.5)) / 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 * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Target

Original	76.0%
Target	99.5%
Herbie	99.8%

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

Derivation

1. Initial program 78.8%

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

2. Simplified 78.8%

   \[\leadsto \color{blue}{\frac{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x}} \]
   Step-by-step derivation

   [Start] 78.8%	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*l* [=>] 78.8%	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
   metadata-eval [=>] 78.8%	\[ \frac{\color{blue}{2.6666666666666665} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x} \]

3. Applied egg-rr 58.0%

   \[\leadsto \frac{\color{blue}{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos x\right)}{2}}}{\sin x} \]
   Step-by-step derivation

   [Start] 78.8%	\[ \frac{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x} \]
   sin-mult [=>] 58.0%	\[ \frac{2.6666666666666665 \cdot \color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x} \]
   associate-*r/ [=>] 58.0%	\[ \frac{\color{blue}{\frac{2.6666666666666665 \cdot \left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right)}{2}}}{\sin x} \]
   +-inverses [=>] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right)}{2}}{\sin x} \]
   cos-sum [=>] 57.9%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \color{blue}{\left(\cos \left(x \cdot 0.5\right) \cdot \cos \left(x \cdot 0.5\right) - \sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}\right)}{2}}{\sin x} \]
   cos-2 [<=] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \color{blue}{\cos \left(2 \cdot \left(x \cdot 0.5\right)\right)}\right)}{2}}{\sin x} \]
   *-commutative [=>] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos \left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right)\right)}{2}}{\sin x} \]
   associate-*r* [=>] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos \color{blue}{\left(\left(2 \cdot 0.5\right) \cdot x\right)}\right)}{2}}{\sin x} \]
   metadata-eval [=>] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos \left(\color{blue}{1} \cdot x\right)\right)}{2}}{\sin x} \]
   *-un-lft-identity [<=] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos \color{blue}{x}\right)}{2}}{\sin x} \]

4. Simplified 58.1%

   \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{0.75}}}{\sin x} \]
   Step-by-step derivation

   [Start] 58.0%	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos 0 - \cos x\right)}{2}}{\sin x} \]
   *-commutative [=>] 58.0%	\[ \frac{\frac{\color{blue}{\left(\cos 0 - \cos x\right) \cdot 2.6666666666666665}}{2}}{\sin x} \]
   associate-/l* [=>] 58.1%	\[ \frac{\color{blue}{\frac{\cos 0 - \cos x}{\frac{2}{2.6666666666666665}}}}{\sin x} \]
   cos-0 [=>] 58.1%	\[ \frac{\frac{\color{blue}{1} - \cos x}{\frac{2}{2.6666666666666665}}}{\sin x} \]
   metadata-eval [=>] 58.1%	\[ \frac{\frac{1 - \cos x}{\color{blue}{0.75}}}{\sin x} \]

5. Taylor expanded in x around inf 58.0%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{1 - \cos x}{\sin x}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{0.75}} \]
   Step-by-step derivation

   [Start] 58.0%	\[ 1.3333333333333333 \cdot \frac{1 - \cos x}{\sin x} \]
   *-commutative [=>] 58.0%	\[ \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 1.3333333333333333} \]
   metadata-eval [<=] 58.0%	\[ \frac{1 - \cos x}{\sin x} \cdot \color{blue}{\frac{1}{0.75}} \]
   times-frac [<=] 58.1%	\[ \color{blue}{\frac{\left(1 - \cos x\right) \cdot 1}{\sin x \cdot 0.75}} \]
   *-rgt-identity [=>] 58.1%	\[ \frac{\color{blue}{1 - \cos x}}{\sin x \cdot 0.75} \]
   associate-/r* [=>] 58.1%	\[ \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{0.75}} \]
   hang-p0-tan [=>] 99.8%	\[ \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{0.75} \]
   div-inv [=>] 99.8%	\[ \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{0.75} \]
   metadata-eval [=>] 99.8%	\[ \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{0.75} \]

7. Final simplification 99.8%

   \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{0.75} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	6720

\[\frac{\tan \left(x \cdot 0.5\right)}{0.75} \]

Alternative 2
Accuracy	99.5%
Cost	6720

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

Alternative 3
Accuracy	51.5%
Cost	704

\[\frac{1}{x \cdot -0.125 + \frac{1}{x} \cdot 1.5} \]

Alternative 4
Accuracy	51.5%
Cost	704

\[\frac{1.3333333333333333}{x \cdot -0.16666666666666666 + 2 \cdot \frac{1}{x}} \]

Alternative 5
Accuracy	51.2%
Cost	320

\[\frac{x \cdot 0.5}{0.75} \]

Alternative 6
Accuracy	50.9%
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2023165 
(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)))