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

Average Error: 14.8 → 0.1

Time: 9.8s

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	14.8
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 14.8

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

2. Simplified 14.8

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

   [Start] 14.8	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*l* [=>] 14.8	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
   associate-*r/ [<=] 14.8	\[ \color{blue}{\frac{8}{3} \cdot \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]
   *-commutative [<=] 14.8	\[ \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8}{3}} \]
   metadata-eval [=>] 14.8	\[ \frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \cdot \color{blue}{2.6666666666666665} \]

3. Applied egg-rr 30.2

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\frac{\cos 0 - \cos x}{0.75}}}} \]

4. Simplified 0.4

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

   [Start] 30.2	\[ \frac{1}{\frac{\sin x}{\frac{\cos 0 - \cos x}{0.75}}} \]
   associate-/r/ [=>] 30.2	\[ \color{blue}{\frac{1}{\sin x} \cdot \frac{\cos 0 - \cos x}{0.75}} \]
   times-frac [<=] 30.2	\[ \color{blue}{\frac{1 \cdot \left(\cos 0 - \cos x\right)}{\sin x \cdot 0.75}} \]
   *-commutative [<=] 30.2	\[ \frac{\color{blue}{\left(\cos 0 - \cos x\right) \cdot 1}}{\sin x \cdot 0.75} \]
   times-frac [=>] 30.2	\[ \color{blue}{\frac{\cos 0 - \cos x}{\sin x} \cdot \frac{1}{0.75}} \]
   cos-0 [=>] 30.2	\[ \frac{\color{blue}{1} - \cos x}{\sin x} \cdot \frac{1}{0.75} \]
   hang-p0-tan [=>] 0.4	\[ \color{blue}{\tan \left(\frac{x}{2}\right)} \cdot \frac{1}{0.75} \]
   metadata-eval [=>] 0.4	\[ \tan \left(\frac{x}{2}\right) \cdot \color{blue}{1.3333333333333333} \]

5. Applied egg-rr 0.4

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

6. Simplified 0.4

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

   [Start] 0.4	\[ \frac{1.3333333333333333 \cdot \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{1.3333333333333333}{\frac{\cos \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)}}} \]

7. Applied egg-rr 0.3

   \[\leadsto \color{blue}{{\left(\frac{0.75}{\tan \left(\frac{x}{2}\right)}\right)}^{-1}} \]

8. Applied egg-rr 0.1

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

9. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.4
Cost	6720

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

Alternative 2
Error	31.1
Cost	832

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

Alternative 3
Error	31.1
Cost	704

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

Alternative 4
Error	31.5
Cost	320

\[\left(x \cdot 0.25\right) \cdot 2.6666666666666665 \]

Alternative 5
Error	31.5
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2023031 
(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)))