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

Average Accuracy: 78.0% → 99.8%

Time: 9.7s

Precision: binary64

Cost: 6720

Math

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

↓

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

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(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	78.0%
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 78.0%

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

2. Simplified 78.0%

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

   [Start] 78.0	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*l* [=>] 78.0	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
   metadata-eval [=>] 78.0	\[ \frac{\color{blue}{2.6666666666666665} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x} \]

3. Applied egg-rr 52.7%

   \[\leadsto \frac{\color{blue}{\frac{2.6666666666666665 \cdot \left(1 - \cos x\right)}{2}}}{\sin x} \]
   Proof

   [Start] 78.0	\[ \frac{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x} \]
   sin-mult [=>] 52.7	\[ \frac{2.6666666666666665 \cdot \color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x} \]
   associate-*r/ [=>] 52.7	\[ \frac{\color{blue}{\frac{2.6666666666666665 \cdot \left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right)}{2}}}{\sin x} \]
   +-inverses [=>] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right)}{2}}{\sin x} \]
   cos-0 [=>] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(\color{blue}{1} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right)}{2}}{\sin x} \]
   cos-sum [=>] 52.4	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \color{blue}{\left(\cos \left(x \cdot 0.5\right) \cdot \cos \left(x \cdot 0.5\right) - \sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}\right)}{2}}{\sin x} \]
   cos-2 [<=] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \color{blue}{\cos \left(2 \cdot \left(x \cdot 0.5\right)\right)}\right)}{2}}{\sin x} \]
   *-commutative [=>] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \cos \left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right)\right)}{2}}{\sin x} \]
   associate-*r* [=>] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \cos \color{blue}{\left(\left(2 \cdot 0.5\right) \cdot x\right)}\right)}{2}}{\sin x} \]
   metadata-eval [=>] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \cos \left(\color{blue}{1} \cdot x\right)\right)}{2}}{\sin x} \]
   *-un-lft-identity [<=] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \cos \color{blue}{x}\right)}{2}}{\sin x} \]

4. Simplified 52.7%

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

   [Start] 52.7	\[ \frac{\frac{2.6666666666666665 \cdot \left(1 - \cos x\right)}{2}}{\sin x} \]
   *-commutative [=>] 52.7	\[ \frac{\frac{\color{blue}{\left(1 - \cos x\right) \cdot 2.6666666666666665}}{2}}{\sin x} \]
   associate-/l* [=>] 52.7	\[ \frac{\color{blue}{\frac{1 - \cos x}{\frac{2}{2.6666666666666665}}}}{\sin x} \]
   metadata-eval [=>] 52.7	\[ \frac{\frac{1 - \cos x}{\color{blue}{0.75}}}{\sin x} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{0 + \frac{\tan \left(\frac{x}{2}\right)}{0.75}} \]
   Proof

   [Start] 52.7	\[ \frac{\frac{1 - \cos x}{0.75}}{\sin x} \]
   add-log-exp [=>] 52.6	\[ \color{blue}{\log \left(e^{\frac{\frac{1 - \cos x}{0.75}}{\sin x}}\right)} \]
   *-un-lft-identity [=>] 52.6	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\frac{1 - \cos x}{0.75}}{\sin x}}\right)} \]
   log-prod [=>] 52.6	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\frac{1 - \cos x}{0.75}}{\sin x}}\right)} \]
   metadata-eval [=>] 52.6	\[ \color{blue}{0} + \log \left(e^{\frac{\frac{1 - \cos x}{0.75}}{\sin x}}\right) \]
   add-log-exp [<=] 52.7	\[ 0 + \color{blue}{\frac{\frac{1 - \cos x}{0.75}}{\sin x}} \]
   associate-/l/ [=>] 52.7	\[ 0 + \color{blue}{\frac{1 - \cos x}{\sin x \cdot 0.75}} \]
   associate-/r* [=>] 52.7	\[ 0 + \color{blue}{\frac{\frac{1 - \cos x}{\sin x}}{0.75}} \]
   hang-p0-tan [=>] 99.8	\[ 0 + \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{0.75} \]

6. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	6720

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

Alternative 2
Accuracy	50.9%
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2023151 
(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)))