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

Percentage Accurate: 76.5% → 99.5%

Time: 10.2s

Precision: binary64

Cost: 19904

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\frac{\frac{\sin x}{t_0}}{2.6666666666666665}} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ t_0 (/ (/ (sin x) t_0) 2.6666666666666665))))

C

double code(double x) {
	return (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
}

↓

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / ((sin(x) / t_0) / 2.6666666666666665);
}

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
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = t_0 / ((sin(x) / t_0) / 2.6666666666666665d0)
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) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 / ((Math.sin(x) / t_0) / 2.6666666666666665);
}

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):
	t_0 = math.sin((x * 0.5))
	return t_0 / ((math.sin(x) / t_0) / 2.6666666666666665)

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)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 / Float64(Float64(sin(x) / t_0) / 2.6666666666666665))
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)
	t_0 = sin((x * 0.5));
	tmp = t_0 / ((sin(x) / t_0) / 2.6666666666666665);
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_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 / N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]]

Target

Original	76.5%
Target	99.5%
Herbie	99.5%

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

Derivation

1. Initial program 74.1%

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

2. Simplified 99.2%

   \[\leadsto \color{blue}{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)} \]
   Step-by-step derivation

   [Start] 74.1%	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*r/ [<=] 99.2%	\[ \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \]
   associate-*l* [=>] 99.2%	\[ \color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)} \]
   metadata-eval [=>] 99.2%	\[ \color{blue}{2.6666666666666665} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right) \]

3. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{2.6666666666666665}}} \]
   Step-by-step derivation

   [Start] 99.2%	\[ 2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right) \]
   associate-*r* [=>] 99.2%	\[ \color{blue}{\left(2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \]
   metadata-eval [<=] 99.2%	\[ \left(\color{blue}{\frac{8}{3}} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x} \]
   associate-*r/ [=>] 74.1%	\[ \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]
   associate-/l* [=>] 99.3%	\[ \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
   metadata-eval [=>] 99.3%	\[ \frac{\color{blue}{2.6666666666666665} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]
   *-commutative [=>] 99.3%	\[ \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \]
   associate-/l* [=>] 99.5%	\[ \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{2.6666666666666665}}} \]

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	19904

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\frac{\frac{\sin x}{t_0}}{2.6666666666666665}} \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	19904

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\sin x} \cdot \frac{t_0}{0.375} \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	19904

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\frac{\sin x}{t_0} \cdot 0.375} \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	6720

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

Alternative 5
Accuracy	51.2%
Cost	192

\[x \cdot 0.6666666666666666 \]

Alternative 6
Accuracy	51.4%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(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)))