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

Average Error: 14.5 → 0.3

Time: 8.2s

Precision: binary64

Cost: 6848

Math

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

↓

\[\frac{1}{\frac{0.75}{\tan \left(\frac{x}{2}\right)}} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 1.0 (/ 0.75 (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 1.0 / (0.75 / 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 = 1.0d0 / (0.75d0 / 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 1.0 / (0.75 / 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 1.0 / (0.75 / 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(1.0 / Float64(0.75 / 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 = 1.0 / (0.75 / 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[(1.0 / N[(0.75 / N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	14.5
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 14.5

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

2. Simplified 14.5

   \[\leadsto \color{blue}{\frac{2.6666666666666665}{\sin x} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)} \]
   Proof
   (*.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
   (*.f64 (/.f64 (Rewrite<= metadata-eval (/.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))): 31 points increase in error, 37 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)): 25 points increase in error, 21 points decrease in error

3. Applied egg-rr 30.2

   \[\leadsto \color{blue}{0 + \frac{2.6666666666666665}{\sin x} \cdot \left(0.5 + -0.5 \cdot \cos x\right)} \]

4. Simplified 30.2

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{1 - \cos x}{\sin x}} \]
   Proof
   (*.f64 4/3 (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (/.f64 8/3 2)) (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 8/3 (-.f64 1 (cos.f64 x))) (*.f64 2 (sin.f64 x)))): 31 points increase in error, 33 points decrease in error
   (/.f64 (*.f64 8/3 (-.f64 1 (cos.f64 x))) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) 2))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 8/3 (sin.f64 x)) (/.f64 (-.f64 1 (cos.f64 x)) 2))): 24 points increase in error, 23 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 1 (cos.f64 x)))) 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 2 (-.f64 1 (cos.f64 x)))))): 21 points increase in error, 5 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 2) (-.f64 1 (cos.f64 x))))): 5 points increase in error, 21 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (*.f64 (Rewrite=> metadata-eval 1/2) (-.f64 1 (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (*.f64 1/2 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 1/2 1) (*.f64 1/2 (neg.f64 (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (+.f64 (Rewrite=> metadata-eval 1/2) (*.f64 1/2 (neg.f64 (cos.f64 x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (+.f64 1/2 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 1/2 (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (+.f64 1/2 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 1/2) (cos.f64 x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 8/3 (sin.f64 x)) (+.f64 1/2 (*.f64 (Rewrite=> metadata-eval -1/2) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 (/.f64 8/3 (sin.f64 x)) (+.f64 1/2 (*.f64 -1/2 (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 30.2

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(1 - \cos x\right) \cdot 1.3333333333333333}}} \]

6. Taylor expanded in x around inf 30.2

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

7. Simplified 0.3

   \[\leadsto \frac{1}{\color{blue}{\frac{0.75}{\tan \left(\frac{x}{2}\right)}}} \]
   Proof
   (/.f64 3/4 (tan.f64 (/.f64 x 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 3/4 (Rewrite<= hang-p0-tan_binary64 (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x)))): 144 points increase in error, 36 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 3/4 (sin.f64 x)) (-.f64 1 (cos.f64 x)))): 28 points increase in error, 22 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 3/4 (/.f64 (sin.f64 x) (-.f64 1 (cos.f64 x))))): 24 points increase in error, 29 points decrease in error

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	6720

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

Alternative 2
Error	31.1
Cost	704

\[\frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \]

Alternative 3
Error	29.6
Cost	704

\[\frac{x \cdot 1.3333333333333333}{1 + \left(1 + x \cdot 0.6666666666666666\right)} \]

Alternative 4
Error	31.5
Cost	320

\[\frac{1}{\frac{1.5}{x}} \]

Alternative 5
Error	31.5
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2022330 
(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)))