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

Average Error: 15.1 → 0.4

Time: 9.2s

Precision: binary64

Cost: 6848

Math

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

↓

\[\frac{2.6666666666666665 \cdot \tan \left(\frac{x}{2}\right)}{2} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (* 2.6666666666666665 (tan (/ x 2.0))) 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 (2.6666666666666665 * tan((x / 2.0))) / 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 = (2.6666666666666665d0 * tan((x / 2.0d0))) / 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 (2.6666666666666665 * Math.tan((x / 2.0))) / 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 (2.6666666666666665 * math.tan((x / 2.0))) / 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(Float64(2.6666666666666665 * tan(Float64(x / 2.0))) / 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 = (2.6666666666666665 * tan((x / 2.0))) / 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[(N[(2.6666666666666665 * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Target

Original	15.1
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 15.1

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

2. Simplified 15.1

   \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]
   Proof
   (*.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (/.f64 8/3 (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))) (/.f64 (Rewrite<= metadata-eval (/.f64 8 3)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (/.f64 8 3) (sin.f64 x)) (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (/.f64 8 3) (*.f64 (sin.f64 (*.f64 x 1/2)) (sin.f64 (*.f64 x 1/2)))) (sin.f64 x))): 39 points increase in error, 34 points decrease in error
   (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 8 3) (sin.f64 (*.f64 x 1/2))) (sin.f64 (*.f64 x 1/2)))) (sin.f64 x)): 26 points increase in error, 28 points decrease in error

3. Applied egg-rr 30.3

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

4. Taylor expanded in x around inf 30.3

   \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot \frac{1 - \cos x}{\sin x}}}{2} \]

5. Simplified 0.4

   \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot \tan \left(\frac{x}{2}\right)}}{2} \]
   Proof
   (*.f64 8/3 (tan.f64 (/.f64 x 2))): 0 points increase in error, 0 points decrease in error
   (*.f64 8/3 (Rewrite<= hang-p0-tan_binary64 (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x)))): 150 points increase in error, 35 points decrease in error

6. Final simplification 0.4

   \[\leadsto \frac{2.6666666666666665 \cdot \tan \left(\frac{x}{2}\right)}{2} \]

Alternatives

Alternative 1
Error	30.1
Cost	584

\[\begin{array}{l} t_0 := 2 + \frac{-6}{x}\\ \mathbf{if}\;x \leq -5.502807647938783 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8512387750046942 \cdot 10^{-18}:\\ \;\;\;\;x \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	29.6
Cost	576

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

Alternative 3
Error	61.7
Cost	192

\[x \cdot -0.6666666666666666 \]

Alternative 4
Error	31.4
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2022317 
(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)))