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

Average Error: 15.0 → 0.1

Time: 9.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

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 0.5)) 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 * 0.5)) / 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 * 0.5d0)) / 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 * 0.5)) / 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 * 0.5)) / 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 * 0.5)) / 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 * 0.5)) / 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 * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Target

Original	15.0
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 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. Simplified 15.0

   \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{2.6666666666666665}{\sin x}} \]
   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))): 38 points increase in error, 29 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 30.2

   \[\leadsto \color{blue}{\frac{1 - \cos x}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]

4. Applied egg-rr 29.6

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	29.3
Cost	968

\[\begin{array}{l} t_0 := \frac{x \cdot 1.3333333333333333}{1 + \left(1 + x \cdot 0.6666666666666666\right)}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.20903663531356512:\\ \;\;\;\;\frac{1}{\frac{1.5}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	29.4
Cost	968

\[\begin{array}{l} t_0 := \frac{x \cdot 1.3333333333333333}{1 + \left(1 + x \cdot 0.6666666666666666\right)}\\ \mathbf{if}\;x \leq -35735.932455582246:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.630164672282824 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	31.5
Cost	320

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

Alternative 4
Error	31.5
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2022308 
(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)))