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

Average Accuracy: 77.1% → 99.7%

Time: 10.0s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ (sin (* x 0.5)) 0.75) (cos (* x 0.5))))

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 (sin((x * 0.5)) / 0.75) / cos((x * 0.5));
}

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 = (sin((x * 0.5d0)) / 0.75d0) / cos((x * 0.5d0))
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.sin((x * 0.5)) / 0.75) / Math.cos((x * 0.5));
}

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.sin((x * 0.5)) / 0.75) / math.cos((x * 0.5))

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(sin(Float64(x * 0.5)) / 0.75) / cos(Float64(x * 0.5)))
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 = (sin((x * 0.5)) / 0.75) / cos((x * 0.5));
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[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision] / N[Cos[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	77.1%
Target	99.5%
Herbie	99.7%

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

Derivation

1. Initial program 77.1%

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

2. Simplified 77.1%

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

   [Start] 77.1	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*l* [=>] 77.1	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
   associate-*r/ [<=] 77.1	\[ \color{blue}{\frac{8}{3} \cdot \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]
   *-commutative [<=] 77.1	\[ \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8}{3}} \]
   metadata-eval [=>] 77.1	\[ \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \color{blue}{2.6666666666666665} \]

3. Applied egg-rr 37.8%

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

   [Start] 77.1	\[ \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot 2.6666666666666665 \]
   expm1-log1p-u [=>] 61.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot 2.6666666666666665\right)\right)} \]
   expm1-udef [=>] 38.7	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot 2.6666666666666665\right)} - 1} \]

4. Simplified 99.4%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333} \]
   Proof

   [Start] 37.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}\right)} - 1 \]
   expm1-def [=>] 37.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}\right)\right)} \]
   expm1-log1p [=>] 53.2	\[ \color{blue}{\frac{\frac{\cos 0 - \cos x}{0.75}}{\sin x}} \]
   associate-/l/ [=>] 53.1	\[ \color{blue}{\frac{\cos 0 - \cos x}{\sin x \cdot 0.75}} \]
   *-commutative [=>] 53.1	\[ \frac{\cos 0 - \cos x}{\color{blue}{0.75 \cdot \sin x}} \]
   metadata-eval [<=] 53.1	\[ \frac{\cos 0 - \cos x}{\color{blue}{\frac{2}{2.6666666666666665}} \cdot \sin x} \]
   associate-/r/ [<=] 53.1	\[ \frac{\cos 0 - \cos x}{\color{blue}{\frac{2}{\frac{2.6666666666666665}{\sin x}}}} \]
   associate-/r/ [=>] 53.1	\[ \color{blue}{\frac{\cos 0 - \cos x}{2} \cdot \frac{2.6666666666666665}{\sin x}} \]
   times-frac [<=] 53.1	\[ \color{blue}{\frac{\left(\cos 0 - \cos x\right) \cdot 2.6666666666666665}{2 \cdot \sin x}} \]
   *-commutative [<=] 53.1	\[ \frac{\left(\cos 0 - \cos x\right) \cdot 2.6666666666666665}{\color{blue}{\sin x \cdot 2}} \]
   times-frac [=>] 53.1	\[ \color{blue}{\frac{\cos 0 - \cos x}{\sin x} \cdot \frac{2.6666666666666665}{2}} \]
   cos-0 [=>] 53.1	\[ \frac{\color{blue}{1} - \cos x}{\sin x} \cdot \frac{2.6666666666666665}{2} \]
   hang-p0-tan [=>] 99.4	\[ \color{blue}{\tan \left(\frac{x}{2}\right)} \cdot \frac{2.6666666666666665}{2} \]
   metadata-eval [=>] 99.4	\[ \tan \left(\frac{x}{2}\right) \cdot \color{blue}{1.3333333333333333} \]

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \left(x \cdot 0.5\right)}} \]
   Proof

   [Start] 99.4	\[ \tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333 \]
   tan-quot [=>] 99.4	\[ \color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}} \cdot 1.3333333333333333 \]
   associate-*l/ [=>] 99.4	\[ \color{blue}{\frac{\sin \left(\frac{x}{2}\right) \cdot 1.3333333333333333}{\cos \left(\frac{x}{2}\right)}} \]
   div-inv [=>] 99.4	\[ \frac{\sin \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot 1.3333333333333333}{\cos \left(\frac{x}{2}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\sin \left(x \cdot \color{blue}{0.5}\right) \cdot 1.3333333333333333}{\cos \left(\frac{x}{2}\right)} \]
   div-inv [=>] 99.4	\[ \frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]
   metadata-eval [=>] 99.4	\[ \frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \left(x \cdot \color{blue}{0.5}\right)} \]

6. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{\sin \left(0.5 \cdot x\right)}{\cos \left(0.5 \cdot x\right)}} \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot 0.5\right)}{0.75}}{\cos \left(x \cdot 0.5\right)}} \]
   Proof

   [Start] 99.4	\[ 1.3333333333333333 \cdot \frac{\sin \left(0.5 \cdot x\right)}{\cos \left(0.5 \cdot x\right)} \]
   *-commutative [<=] 99.4	\[ 1.3333333333333333 \cdot \frac{\sin \color{blue}{\left(x \cdot 0.5\right)}}{\cos \left(0.5 \cdot x\right)} \]
   *-commutative [<=] 99.4	\[ 1.3333333333333333 \cdot \frac{\sin \left(x \cdot 0.5\right)}{\cos \color{blue}{\left(x \cdot 0.5\right)}} \]
   metadata-eval [<=] 99.4	\[ \color{blue}{\frac{1}{0.75}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\cos \left(x \cdot 0.5\right)} \]
   times-frac [<=] 99.7	\[ \color{blue}{\frac{1 \cdot \sin \left(x \cdot 0.5\right)}{0.75 \cdot \cos \left(x \cdot 0.5\right)}} \]
   *-lft-identity [=>] 99.7	\[ \frac{\color{blue}{\sin \left(x \cdot 0.5\right)}}{0.75 \cdot \cos \left(x \cdot 0.5\right)} \]
   associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{\sin \left(x \cdot 0.5\right)}{0.75}}{\cos \left(x \cdot 0.5\right)}} \]

8. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	6720

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

Alternative 2
Accuracy	51.0%
Cost	704

\[\frac{2.6666666666666665}{x \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x}} \]

Alternative 3
Accuracy	50.5%
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2023135 
(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)))