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

Percentage Accurate: 76.9% → 99.8%

Time: 23.5s

Alternatives: 7

Speedup: 3.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t_0\right) \cdot t_0}{\sin x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t_0\right) \cdot t_0}{\sin x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\tan \left(\frac{x}{2}\right)}{0.75} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (tan (/ x 2.0)) 0.75))

C

double code(double x) {
	return tan((x / 2.0)) / 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) / 0.75d0
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) / 0.75;
}

Python

def code(x):
	return math.tan((x / 2.0)) / 0.75

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) / 0.75)
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) / 0.75;
end

Wolfram

code[x_] := N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. *-lft-identity 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

   3. metadata-eval 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

   4. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   5. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   7. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   8. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   9. distribute-lft-neg-out 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   10. associate-/l* 99.2%

       \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   11. associate-*r/ 99.3%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.3%

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

   1. associate-/l* 80.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-/l* 80.4%

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

   5. sin-mult 58.4%

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

   6. associate-/l/ 58.4%

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

   7. div-inv 58.4%

      \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right) \cdot \frac{1}{\frac{\sin x}{2.6666666666666665} \cdot 2}} \]

5. Applied egg-rr 58.5%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
6. Step-by-step derivation

   1. un-div-inv 58.5%

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

   2. associate-*l* 58.5%

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

   3. associate-/r* 58.4%

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

   4. hang-p0-tan 99.7%

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

   5. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 58.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \lor \neg \left(x \leq 3.1\right):\\ \;\;\;\;\sin x \cdot 2.6666666666666665\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.2) (not (<= x 3.1)))
   (* (sin x) 2.6666666666666665)
   (/ 1.0 (+ (* x -0.125) (* 1.5 (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if ((x <= -3.2) || !(x <= 3.1)) {
		tmp = sin(x) * 2.6666666666666665;
	} else {
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.2d0)) .or. (.not. (x <= 3.1d0))) then
        tmp = sin(x) * 2.6666666666666665d0
    else
        tmp = 1.0d0 / ((x * (-0.125d0)) + (1.5d0 * (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -3.2) || !(x <= 3.1)) {
		tmp = Math.sin(x) * 2.6666666666666665;
	} else {
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.2) or not (x <= 3.1):
		tmp = math.sin(x) * 2.6666666666666665
	else:
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.2) || !(x <= 3.1))
		tmp = Float64(sin(x) * 2.6666666666666665);
	else
		tmp = Float64(1.0 / Float64(Float64(x * -0.125) + Float64(1.5 * Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -3.2) || ~((x <= 3.1)))
		tmp = sin(x) * 2.6666666666666665;
	else
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.2], N[Not[LessEqual[x, 3.1]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * 2.6666666666666665), $MachinePrecision], N[(1.0 / N[(N[(x * -0.125), $MachinePrecision] + N[(1.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002 or 3.10000000000000009 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. *-commutative 99.1%

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

      2. *-lft-identity 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

      3. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

      4. times-frac 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

      5. associate-/l* 99.0%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      6. *-commutative 99.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      7. neg-mul-1 99.0%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      8. sin-neg 99.0%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      9. distribute-lft-neg-out 99.0%

         \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      10. associate-/l* 99.0%

          \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      11. associate-*r/ 99.1%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.1%

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

      1. associate-/l* 99.0%

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

      2. clear-num 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}{2.6666666666666665}}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}} \cdot 2.6666666666666665} \]

   5. Applied egg-rr 18.4%

      \[\leadsto \color{blue}{\sin x \cdot 2.6666666666666665} \]

   if -3.2000000000000002 < x < 3.10000000000000009

   1. Initial program 56.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. *-commutative 56.3%

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

      2. *-lft-identity 56.3%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

      3. metadata-eval 56.3%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

      4. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

      5. associate-/l* 99.6%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      6. *-commutative 99.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      7. neg-mul-1 99.6%

         \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      8. sin-neg 99.6%

         \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      9. distribute-lft-neg-out 99.6%

         \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

      10. associate-/l* 99.6%

          \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

      11. associate-*r/ 99.5%

          \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.5%

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

      1. associate-/l* 56.3%

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

      2. associate-*l* 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-/l* 56.2%

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

      5. sin-mult 6.9%

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

      6. associate-/l/ 6.9%

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

      7. div-inv 6.9%

         \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right) \cdot \frac{1}{\frac{\sin x}{2.6666666666666665} \cdot 2}} \]

   5. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
   6. Step-by-step derivation

      1. un-div-inv 6.9%

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

      2. associate-*l* 6.9%

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

      3. associate-/r* 6.9%

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

      4. clear-num 6.9%

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

      5. metadata-eval 6.9%

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

      6. hang-p0-tan 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 99.5%

      \[\leadsto \frac{1}{\color{blue}{-0.125 \cdot x + 1.5 \cdot \frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \lor \neg \left(x \leq 3.1\right):\\ \;\;\;\;\sin x \cdot 2.6666666666666665\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (tan (/ x 2.0)) 1.3333333333333333))

C

double code(double x) {
	return tan((x / 2.0)) * 1.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0)) * 1.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0)) * 1.3333333333333333;
}

Python

def code(x):
	return math.tan((x / 2.0)) * 1.3333333333333333

Julia

function code(x)
	return Float64(tan(Float64(x / 2.0)) * 1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0)) * 1.3333333333333333;
end

Wolfram

code[x_] := N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. *-lft-identity 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

   3. metadata-eval 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

   4. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   5. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   7. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   8. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   9. distribute-lft-neg-out 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   10. associate-/l* 99.2%

       \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   11. associate-*r/ 99.3%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.3%

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

   1. associate-/l* 80.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-/l* 80.4%

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

   5. sin-mult 58.4%

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

   6. associate-/l/ 58.4%

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

   7. div-inv 58.4%

      \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right) \cdot \frac{1}{\frac{\sin x}{2.6666666666666665} \cdot 2}} \]

5. Applied egg-rr 58.5%

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

6. Taylor expanded in x around inf 58.5%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{1 - \cos x}{\sin x}} \]
7. Step-by-step derivation

   1. hang-p0-tan 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 51.5% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (* x -0.125) (* 1.5 (/ 1.0 x)))))

C

double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x * (-0.125d0)) + (1.5d0 * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Python

def code(x):
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x * -0.125) + Float64(1.5 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * -0.125), $MachinePrecision] + N[(1.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. *-lft-identity 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

   3. metadata-eval 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

   4. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   5. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   7. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   8. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   9. distribute-lft-neg-out 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   10. associate-/l* 99.2%

       \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   11. associate-*r/ 99.3%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.3%

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

   1. associate-/l* 80.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-/l* 80.4%

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

   5. sin-mult 58.4%

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

   6. associate-/l/ 58.4%

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

   7. div-inv 58.4%

      \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right) \cdot \frac{1}{\frac{\sin x}{2.6666666666666665} \cdot 2}} \]

5. Applied egg-rr 58.5%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
6. Step-by-step derivation

   1. un-div-inv 58.5%

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

   2. associate-*l* 58.5%

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

   3. associate-/r* 58.4%

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

   4. clear-num 58.4%

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

   5. metadata-eval 58.4%

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

   6. hang-p0-tan 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in x around 0 45.7%

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

9. Final simplification 45.7%

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

Alternative 5: 51.1% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.5 x)))

C

double code(double x) {
	return 1.0 / (1.5 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.5d0 / x)
end function

Java

public static double code(double x) {
	return 1.0 / (1.5 / x);
}

Python

def code(x):
	return 1.0 / (1.5 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.5 / x);
end

Wolfram

code[x_] := N[(1.0 / N[(1.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. *-lft-identity 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

   3. metadata-eval 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

   4. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   5. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   7. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   8. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   9. distribute-lft-neg-out 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   10. associate-/l* 99.2%

       \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   11. associate-*r/ 99.3%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.3%

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

   1. associate-/l* 80.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-/l* 80.4%

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

   5. sin-mult 58.4%

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

   6. associate-/l/ 58.4%

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

   7. div-inv 58.4%

      \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right) \cdot \frac{1}{\frac{\sin x}{2.6666666666666665} \cdot 2}} \]

5. Applied egg-rr 58.5%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
6. Step-by-step derivation

   1. un-div-inv 58.5%

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

   2. associate-*l* 58.5%

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

   3. associate-/r* 58.4%

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

   4. clear-num 58.4%

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

   5. metadata-eval 58.4%

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

   6. hang-p0-tan 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in x around 0 45.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]

9. Final simplification 45.3%

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

Alternative 6: 51.3% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.5}{0.75} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 0.5) 0.75))

C

double code(double x) {
	return (x * 0.5) / 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 0.5d0) / 0.75d0
end function

Java

public static double code(double x) {
	return (x * 0.5) / 0.75;
}

Python

def code(x):
	return (x * 0.5) / 0.75

Julia

function code(x)
	return Float64(Float64(x * 0.5) / 0.75)
end

MATLAB

function tmp = code(x)
	tmp = (x * 0.5) / 0.75;
end

Wolfram

code[x_] := N[(N[(x * 0.5), $MachinePrecision] / 0.75), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. *-lft-identity 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

   3. metadata-eval 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

   4. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   5. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   7. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   8. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   9. distribute-lft-neg-out 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   10. associate-/l* 99.2%

       \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   11. associate-*r/ 99.3%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.3%

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

   1. associate-/l* 80.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-/l* 80.4%

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

   5. sin-mult 58.4%

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

   6. associate-/l/ 58.4%

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

   7. div-inv 58.4%

      \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)\right) \cdot \frac{1}{\frac{\sin x}{2.6666666666666665} \cdot 2}} \]

5. Applied egg-rr 58.5%

   \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
6. Step-by-step derivation

   1. un-div-inv 58.5%

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

   2. associate-*l* 58.5%

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

   3. associate-/r* 58.4%

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

   4. hang-p0-tan 99.7%

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

   5. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0 45.4%

   \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{0.75} \]
9. Step-by-step derivation

   1. *-commutative 45.4%

      \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

10. Simplified 45.4%

    \[\leadsto \frac{\color{blue}{x \cdot 0.5}}{0.75} \]

11. Final simplification 45.4%

    \[\leadsto \frac{x \cdot 0.5}{0.75} \]

Alternative 7: 51.0% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.6666666666666666))

C

double code(double x) {
	return x * 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return x * 0.6666666666666666;
}

Python

def code(x):
	return x * 0.6666666666666666

Julia

function code(x)
	return Float64(x * 0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.6666666666666666;
end

Wolfram

code[x_] := N[(x * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. *-commutative 80.4%

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

   2. *-lft-identity 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}} \]

   3. metadata-eval 80.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{\frac{-1}{-1}} \cdot \sin x} \]

   4. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   5. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   7. neg-mul-1 99.2%

      \[\leadsto \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   8. sin-neg 99.2%

      \[\leadsto \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   9. distribute-lft-neg-out 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \cdot \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   10. associate-/l* 99.2%

       \[\leadsto \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   11. associate-*r/ 99.3%

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1} \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 45.2%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 45.2%

      \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

6. Simplified 45.2%

   \[\leadsto \color{blue}{x \cdot 0.6666666666666666} \]

7. Final simplification 45.2%

   \[\leadsto x \cdot 0.6666666666666666 \]

Developer target: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\frac{8 \cdot t_0}{3}}{\frac{\sin x}{t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (/ (* 8.0 t_0) 3.0) (/ (sin x) t_0))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = ((8.0d0 * t_0) / 3.0d0) / (sin(x) / t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return ((8.0 * t_0) / 3.0) / (Math.sin(x) / t_0);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return ((8.0 * t_0) / 3.0) / (math.sin(x) / t_0)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(8.0 * t_0) / 3.0) / Float64(sin(x) / t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(8.0 * t$95$0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023297 
(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)))