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

Average Error: 15.0 → 0.4

Time: 8.9s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (* 1.3333333333333333 (tan (/ x 2.0))))

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 1.3333333333333333 * tan((x / 2.0));
}

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 = 1.3333333333333333d0 * tan((x / 2.0d0))
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 1.3333333333333333 * Math.tan((x / 2.0));
}

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 1.3333333333333333 * math.tan((x / 2.0))

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(1.3333333333333333 * tan(Float64(x / 2.0)))
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 = 1.3333333333333333 * tan((x / 2.0));
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[(1.3333333333333333 * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	15.0
Target	0.3
Herbie	0.4

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

Derivation

1. Initial program 15.0

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

2. Applied egg-rr 30.1

   \[\leadsto \color{blue}{{\left(0.375 \cdot \frac{\sin x}{0.5 - 0.5 \cdot \cos x}\right)}^{-1}} \]

3. Taylor expanded in x around inf 30.1

   \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}} \]

4. Simplified 0.4

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

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	29.5
Cost	584

\[\begin{array}{l} t_0 := 2 + \frac{-6}{x}\\ \mathbf{if}\;x \leq -3828738.1032977384:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4746593907131898 \cdot 10^{-7}:\\ \;\;\;\;x \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	29.4
Cost	584

\[\begin{array}{l} t_0 := 2 + \frac{-6}{x}\\ \mathbf{if}\;x \leq -3828738.1032977384:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4746593907131898 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	29.6
Cost	576

\[\frac{1.3333333333333333 \cdot x}{2 + x \cdot 0.6666666666666666} \]

Alternative 4
Error	31.5
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2022228 
(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)))