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

Average Error: 14.8 → 0.3

Time: 27.7s

Precision: binary64

Cost: 19976

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t_1}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{\sin x}}{0.375}\\ \end{array} \]

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_1 (pow t_0 2.0)))
   (if (<= x -5e-5)
     (/ (/ t_1 0.375) (sin x))
     (if (<= x 5e-42) (/ t_0 0.75) (/ (/ t_1 (sin x)) 0.375)))))

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));
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (x <= -5e-5) {
		tmp = (t_1 / 0.375) / sin(x);
	} else if (x <= 5e-42) {
		tmp = t_0 / 0.75;
	} else {
		tmp = (t_1 / sin(x)) / 0.375;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    t_1 = t_0 ** 2.0d0
    if (x <= (-5d-5)) then
        tmp = (t_1 / 0.375d0) / sin(x)
    else if (x <= 5d-42) then
        tmp = t_0 / 0.75d0
    else
        tmp = (t_1 / sin(x)) / 0.375d0
    end if
    code = tmp
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));
	double t_1 = Math.pow(t_0, 2.0);
	double tmp;
	if (x <= -5e-5) {
		tmp = (t_1 / 0.375) / Math.sin(x);
	} else if (x <= 5e-42) {
		tmp = t_0 / 0.75;
	} else {
		tmp = (t_1 / Math.sin(x)) / 0.375;
	}
	return tmp;
}

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))
	t_1 = math.pow(t_0, 2.0)
	tmp = 0
	if x <= -5e-5:
		tmp = (t_1 / 0.375) / math.sin(x)
	elif x <= 5e-42:
		tmp = t_0 / 0.75
	else:
		tmp = (t_1 / math.sin(x)) / 0.375
	return tmp

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))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -5e-5)
		tmp = Float64(Float64(t_1 / 0.375) / sin(x));
	elseif (x <= 5e-42)
		tmp = Float64(t_0 / 0.75);
	else
		tmp = Float64(Float64(t_1 / sin(x)) / 0.375);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x);
end

↓

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	t_1 = t_0 ^ 2.0;
	tmp = 0.0;
	if (x <= -5e-5)
		tmp = (t_1 / 0.375) / sin(x);
	elseif (x <= 5e-42)
		tmp = t_0 / 0.75;
	else
		tmp = (t_1 / sin(x)) / 0.375;
	end
	tmp_2 = tmp;
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]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -5e-5], N[(N[(t$95$1 / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-42], N[(t$95$0 / 0.75), $MachinePrecision], N[(N[(t$95$1 / N[Sin[x], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]]]]

Target

Original	14.8
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. Split input into 3 regimes

2. if x < -5.00000000000000024e-5

   1. Initial program 0.6

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

   2. Simplified 0.7

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

      [Start] 0.6	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
      rational.json-simplify-2 [=>] 0.6	\[ \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
      rational.json-simplify-43 [=>] 0.6	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
      rational.json-simplify-49 [=>] 0.7	\[ \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\frac{8}{3}}{\sin x}} \]
      metadata-eval [=>] 0.7	\[ \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\color{blue}{2.6666666666666665}}{\sin x} \]

   3. Taylor expanded in x around inf 0.6

      \[\leadsto \color{blue}{{\sin \left(0.5 \cdot x\right)}^{2}} \cdot \frac{2.6666666666666665}{\sin x} \]

   4. Simplified 0.6

      \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \frac{2.6666666666666665}{\sin x} \]
      Proof

      [Start] 0.6	\[ {\sin \left(0.5 \cdot x\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x} \]
      rational.json-simplify-2 [<=] 0.6	\[ {\sin \color{blue}{\left(x \cdot 0.5\right)}}^{2} \cdot \frac{2.6666666666666665}{\sin x} \]

   5. Applied egg-rr 0.6

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{\sin x}} \]

   if -5.00000000000000024e-5 < x < 5.00000000000000003e-42

   1. Initial program 31.1

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

   2. Simplified 0.3

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

      [Start] 31.1	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
      rational.json-simplify-2 [=>] 31.1	\[ \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
      rational.json-simplify-49 [=>] 0.3	\[ \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \]
      rational.json-simplify-2 [=>] 0.3	\[ \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)} \]
      rational.json-simplify-43 [=>] 0.3	\[ \color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)} \]
      metadata-eval [=>] 0.3	\[ \color{blue}{2.6666666666666665} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right) \]

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{x}{-2}\right)}{\frac{-\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-0.375}{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}}} \]
      Proof

      [Start] 0.4	\[ \frac{\sin \left(\frac{x}{-2}\right)}{\frac{-\sin x}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}} \]
      rational.json-simplify-61 [=>] 0.3	\[ \color{blue}{\frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{-\sin x}{\sin \left(\frac{x}{-2}\right)}}} \]
      rational.json-simplify-10 [=>] 0.3	\[ \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{\frac{\sin x}{-1}}}{\sin \left(\frac{x}{-2}\right)}} \]
      rational.json-simplify-44 [=>] 0.3	\[ \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{\frac{\sin x}{\sin \left(\frac{x}{-2}\right)}}{-1}}} \]
      rational.json-simplify-47 [=>] 0.3	\[ \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{\sin x}{\sin \left(\frac{x}{-2}\right) \cdot -1}}} \]
      rational.json-simplify-8 [<=] 0.3	\[ \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{-\sin \left(\frac{x}{-2}\right)}}} \]
      rational.json-simplify-11 [<=] 0.3	\[ \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\color{blue}{\frac{\sin \left(\frac{x}{-2}\right)}{-1}}}} \]
      rational.json-simplify-61 [<=] 0.3	\[ \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{-1}{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}}} \]
      rational.json-simplify-61 [<=] 0.4	\[ \color{blue}{\frac{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}{\frac{-1}{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}}} \]
      rational.json-simplify-46 [=>] 0.3	\[ \frac{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}{\color{blue}{\frac{\frac{-1}{2.6666666666666665}}{\sin \left(x \cdot 0.5\right)}}} \]
      rational.json-simplify-61 [=>] 0.0	\[ \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\frac{-1}{2.6666666666666665}}{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}}} \]
      metadata-eval [=>] 0.0	\[ \frac{\sin \left(x \cdot 0.5\right)}{\frac{\color{blue}{-0.375}}{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}} \]

   5. Taylor expanded in x around 0 0.1

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

   if 5.00000000000000003e-42 < x

   1. Initial program 0.6

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

   2. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
      rational.json-simplify-2 [=>] 0.6	\[ \frac{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
      rational.json-simplify-43 [=>] 0.6	\[ \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)}}{\sin x} \]
      rational.json-simplify-49 [=>] 0.6	\[ \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\frac{8}{3}}{\sin x}} \]
      metadata-eval [=>] 0.6	\[ \left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\color{blue}{2.6666666666666665}}{\sin x} \]

   3. Taylor expanded in x around inf 0.6

      \[\leadsto \color{blue}{{\sin \left(0.5 \cdot x\right)}^{2}} \cdot \frac{2.6666666666666665}{\sin x} \]

   4. Simplified 0.6

      \[\leadsto \color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \frac{2.6666666666666665}{\sin x} \]
      Proof

      [Start] 0.6	\[ {\sin \left(0.5 \cdot x\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x} \]
      rational.json-simplify-2 [<=] 0.6	\[ {\sin \color{blue}{\left(x \cdot 0.5\right)}}^{2} \cdot \frac{2.6666666666666665}{\sin x} \]

   5. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	19976

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := \frac{{t_0}^{2}}{\sin x} \cdot 2.6666666666666665\\ \mathbf{if}\;x \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-15}:\\ \;\;\;\;\frac{t_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.3
Cost	19976

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t_1}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 10^{-15}:\\ \;\;\;\;\frac{t_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sin x} \cdot 2.6666666666666665\\ \end{array} \]

Alternative 3
Error	0.3
Cost	19904

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

Alternative 4
Error	0.5
Cost	19904

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

Alternative 5
Error	0.4
Cost	19904

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

Alternative 6
Error	0.3
Cost	19904

\[\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot -0.5\right)} \cdot -0.375} \]

Alternative 7
Error	0.3
Cost	19904

\[\frac{\sin \left(x \cdot 0.5\right)}{\frac{-0.375}{\frac{\sin \left(\frac{x}{-2}\right)}{\sin x}}} \]

Alternative 8
Error	28.7
Cost	6720

\[\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 \]

Alternative 9
Error	28.5
Cost	6720

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

Alternative 10
Error	31.0
Cost	704

\[\frac{2.6666666666666665}{x \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x}} \]

Alternative 11
Error	31.1
Cost	320

\[4 \cdot \frac{x}{6} \]

Alternative 12
Error	31.3
Cost	192

\[x \cdot 0.6666666666666666 \]

Reproduce

herbie shell --seed 2023064 
(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)))