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

Percentage Accurate: 76.8% → 99.5%

Time: 11.6s

Alternatives: 15

Speedup: 1.0×

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: 76.8% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t\_0 \cdot \frac{t\_0}{-\sin x}}{-0.375} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* t_0 (/ t_0 (- (sin x)))) -0.375)))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (t_0 * (t_0 / -sin(x))) / -0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (t_0 * (t_0 / -sin(x))) / (-0.375d0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (t_0 * (t_0 / -Math.sin(x))) / -0.375;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (t_0 * (t_0 / -math.sin(x))) / -0.375

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(t_0 * Float64(t_0 / Float64(-sin(x)))) / -0.375)
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (t_0 * (t_0 / -sin(x))) / -0.375;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(t$95$0 / (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / -0.375), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.2%

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

   6. un-div-inv 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. times-frac 99.6%

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

   9. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around inf 99.6%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.375 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*r/ 99.5%

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

9. Simplified 99.5%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\sin x \cdot \frac{0.375}{\sin \left(x \cdot 0.5\right)}}} \]
10. Step-by-step derivation

    1. associate-/r* 99.4%

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

    2. associate-/r/ 99.4%

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

11. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}{0.375} \cdot \sin \left(x \cdot 0.5\right)} \]
12. Step-by-step derivation

    1. frac-2neg 99.4%

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

    2. associate-*l/ 99.6%

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

    3. distribute-neg-frac2 99.6%

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

    4. metadata-eval 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{-\sin x}}{-0.375} \]
15. Add Preprocessing

Alternative 2: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0001:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0001)
   (/ x (* 0.375 (fma (pow x 2.0) -0.3333333333333333 4.0)))
   (/ 2.6666666666666665 (* (sin x) (pow (sin (* x 0.5)) -2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0001) {
		tmp = x / (0.375 * fma(pow(x, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 2.6666666666666665 / (sin(x) * pow(sin((x * 0.5)), -2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0001)
		tmp = Float64(x / Float64(0.375 * fma((x ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x) * (sin(Float64(x * 0.5)) ^ -2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0001], N[(x / N[(0.375 * N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] * N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 69.2%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. associate-*r/ 69.2%

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

      3. metadata-eval 69.2%

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

      4. clear-num 69.2%

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

      5. *-un-lft-identity 69.2%

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

      6. metadata-eval 69.2%

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

      7. associate-*l* 69.2%

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

      8. times-frac 69.2%

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

      9. metadata-eval 69.2%

         \[\leadsto \frac{1}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

      10. pow2 69.2%

          \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. Taylor expanded in x around 0 68.2%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 68.2%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 68.2%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 68.2%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{0.375 \cdot \frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. associate-/r* 68.3%

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       3. metadata-eval 68.3%

          \[\leadsto 1 \cdot \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       4. div-inv 68.3%

          \[\leadsto 1 \cdot \color{blue}{\left(2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}\right)} \]

       5. clear-num 68.3%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}}\right) \]

       6. +-commutative 68.3%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}}\right) \]

       7. fma-define 68.3%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}}\right) \]

   11. Applied egg-rr 68.3%

       \[\leadsto \color{blue}{1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 68.3%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       2. metadata-eval 68.3%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       3. times-frac 68.6%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       4. *-lft-identity 68.6%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 68.6%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 1.00000000000000005e-4 < x

   1. Initial program 98.8%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. associate-*r/ 98.9%

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

      2. clear-num 98.9%

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

      3. pow2 98.9%

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

   6. Applied egg-rr 98.9%

      \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{1}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. un-div-inv 98.9%

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

      2. div-inv 99.0%

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

      3. pow-flip 99.2%

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

      4. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t\_0}^{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (<= x 2e-9)
     (/ t_0 0.75)
     (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if (x <= 2e-9) {
		tmp = t_0 / 0.75;
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    if (x <= 2d-9) then
        tmp = t_0 / 0.75d0
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if (x <= 2e-9) {
		tmp = t_0 / 0.75;
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if x <= 2e-9:
		tmp = t_0 / 0.75
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if (x <= 2e-9)
		tmp = Float64(t_0 / 0.75);
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if (x <= 2e-9)
		tmp = t_0 / 0.75;
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2e-9], N[(t$95$0 / 0.75), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000012e-9

   1. Initial program 69.0%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. div-inv 99.2%

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

      4. associate-*l* 99.1%

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

      5. associate-/r/ 99.2%

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

      6. un-div-inv 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. times-frac 99.7%

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

      9. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 70.8%

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

   if 2.00000000000000012e-9 < x

   1. Initial program 98.9%

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

      1. metadata-eval 98.9%

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

      2. associate-*r/ 98.9%

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

      3. associate-*r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r/ 98.9%

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

      6. pow2 98.9%

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

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x} \cdot 2.6666666666666665} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t\_0}{0.375 \cdot \frac{\sin x}{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ t_0 (* 0.375 (/ (sin x) t_0)))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / (0.375 * (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 = t_0 / (0.375d0 * (sin(x) / t_0))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 / (0.375 * (Math.sin(x) / t_0));
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 / (0.375 * (math.sin(x) / t_0))

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 / Float64(0.375 * Float64(sin(x) / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 / (0.375 * (sin(x) / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 / N[(0.375 * N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.2%

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

   6. un-div-inv 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. times-frac 99.6%

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

   9. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
7. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t\_0 \cdot \frac{\frac{t\_0}{\sin x}}{0.375} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (* t_0 (/ (/ t_0 (sin x)) 0.375))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 * ((t_0 / sin(x)) / 0.375);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = t_0 * ((t_0 / sin(x)) / 0.375d0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 * ((t_0 / Math.sin(x)) / 0.375);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 * ((t_0 / math.sin(x)) / 0.375)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 * Float64(Float64(t_0 / sin(x)) / 0.375))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 * ((t_0 / sin(x)) / 0.375);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.2%

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

   6. un-div-inv 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. times-frac 99.6%

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

   9. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around inf 99.6%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.375 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*r/ 99.5%

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

9. Simplified 99.5%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\sin x \cdot \frac{0.375}{\sin \left(x \cdot 0.5\right)}}} \]
10. Step-by-step derivation

    1. associate-/r* 99.4%

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

    2. associate-/r/ 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}{0.375} \]
13. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t\_0 \cdot \frac{2.6666666666666665}{\frac{\sin x}{t\_0}} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 * (2.6666666666666665 / (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 = t_0 * (2.6666666666666665d0 / (sin(x) / t_0))
end function

Java

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

Python

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

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 * Float64(2.6666666666666665 / Float64(sin(x) / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 * (2.6666666666666665 / (sin(x) / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.2%

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

   2. inv-pow 99.2%

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

   3. *-un-lft-identity 99.2%

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

   4. times-frac 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot {\left(\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{-1} \]

6. Applied egg-rr 99.4%

   \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{{\left(0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 99.4%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   2. associate-/r* 99.3%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\frac{1}{0.375}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. metadata-eval 99.3%

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

8. Simplified 99.3%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t\_0 \cdot \left(t\_0 \cdot \frac{2.6666666666666665}{\sin x}\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = t_0 * (t_0 * (2.6666666666666665d0 / sin(x)))
end function

Java

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

Python

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

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 * Float64(t_0 * Float64(2.6666666666666665 / sin(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 * (t_0 * (2.6666666666666665 / sin(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*r/ 99.3%

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

7. Simplified 99.3%

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

Alternative 8: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ 2.6666666666666665 \cdot \left(t\_0 \cdot \frac{t\_0}{\sin x}\right) \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return 2.6666666666666665 * (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 = 2.6666666666666665d0 * (t_0 * (t_0 / sin(x)))
end function

Java

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

Python

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

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(2.6666666666666665 * Float64(t_0 * Float64(t_0 / sin(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = 2.6666666666666665 * (t_0 * (t_0 / sin(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(2.6666666666666665 * N[(t$95$0 * N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos x}{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (/ x (* 0.375 (fma (pow x 2.0) -0.3333333333333333 4.0)))
   (/ 1.0 (* 0.375 (/ (sin x) (- 0.5 (/ (cos x) 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = x / (0.375 * fma(pow(x, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 1.0 / (0.375 * (sin(x) / (0.5 - (cos(x) / 2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(x / Float64(0.375 * fma((x ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(1.0 / Float64(0.375 * Float64(sin(x) / Float64(0.5 - Float64(cos(x) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(x / N[(0.375 * N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.375 * N[(N[Sin[x], $MachinePrecision] / N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 69.3%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. associate-*r/ 69.3%

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

      3. metadata-eval 69.3%

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

      4. clear-num 69.3%

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

      5. *-un-lft-identity 69.3%

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

      6. metadata-eval 69.3%

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

      7. associate-*l* 69.4%

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

      8. times-frac 69.4%

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

      9. metadata-eval 69.4%

         \[\leadsto \frac{1}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

      10. pow2 69.4%

          \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   6. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. Taylor expanded in x around 0 68.3%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 68.3%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 68.3%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 68.3%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{0.375 \cdot \frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. associate-/r* 68.4%

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       3. metadata-eval 68.4%

          \[\leadsto 1 \cdot \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       4. div-inv 68.4%

          \[\leadsto 1 \cdot \color{blue}{\left(2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}\right)} \]

       5. clear-num 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}}\right) \]

       6. +-commutative 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}}\right) \]

       7. fma-define 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}}\right) \]

   11. Applied egg-rr 68.4%

       \[\leadsto \color{blue}{1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 68.4%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       2. metadata-eval 68.4%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       3. times-frac 68.7%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       4. *-lft-identity 68.7%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 0.00419999999999999974 < x

   1. Initial program 98.8%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. associate-*r* 98.9%

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

      2. associate-*r/ 98.8%

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

      3. metadata-eval 98.8%

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

      4. clear-num 98.9%

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

      5. *-un-lft-identity 98.9%

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

      6. metadata-eval 98.9%

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

      7. associate-*l* 98.8%

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

      8. times-frac 98.9%

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

      9. metadata-eval 98.9%

         \[\leadsto \frac{1}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

      10. pow2 98.9%

          \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   6. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 98.9%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

      2. sin-mult 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]

   8. Applied egg-rr 98.0%

      \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]
   9. Step-by-step derivation

      1. div-sub 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]

      2. +-inverses 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      3. cos-0 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      4. metadata-eval 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      5. distribute-lft-out 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}} \]

      6. metadata-eval 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}} \]

      7. *-rgt-identity 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \color{blue}{x}}{2}}} \]

   10. Simplified 98.0%

       \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{0.5 - \frac{\cos x}{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{1}{\frac{\sin x}{0.5 - \frac{\cos x}{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (/ x (* 0.375 (fma (pow x 2.0) -0.3333333333333333 4.0)))
   (* 2.6666666666666665 (/ 1.0 (/ (sin x) (- 0.5 (/ (cos x) 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = x / (0.375 * fma(pow(x, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 2.6666666666666665 * (1.0 / (sin(x) / (0.5 - (cos(x) / 2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(x / Float64(0.375 * fma((x ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(2.6666666666666665 * Float64(1.0 / Float64(sin(x) / Float64(0.5 - Float64(cos(x) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(x / N[(0.375 * N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(1.0 / N[(N[Sin[x], $MachinePrecision] / N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 69.3%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. associate-*r/ 69.3%

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

      3. metadata-eval 69.3%

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

      4. clear-num 69.3%

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

      5. *-un-lft-identity 69.3%

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

      6. metadata-eval 69.3%

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

      7. associate-*l* 69.4%

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

      8. times-frac 69.4%

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

      9. metadata-eval 69.4%

         \[\leadsto \frac{1}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

      10. pow2 69.4%

          \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   6. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. Taylor expanded in x around 0 68.3%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 68.3%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 68.3%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 68.3%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{0.375 \cdot \frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. associate-/r* 68.4%

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       3. metadata-eval 68.4%

          \[\leadsto 1 \cdot \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       4. div-inv 68.4%

          \[\leadsto 1 \cdot \color{blue}{\left(2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}\right)} \]

       5. clear-num 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}}\right) \]

       6. +-commutative 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}}\right) \]

       7. fma-define 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}}\right) \]

   11. Applied egg-rr 68.4%

       \[\leadsto \color{blue}{1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 68.4%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       2. metadata-eval 68.4%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       3. times-frac 68.7%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       4. *-lft-identity 68.7%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 0.00419999999999999974 < x

   1. Initial program 98.8%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. associate-*r/ 98.8%

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

      2. clear-num 98.9%

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

      3. pow2 98.9%

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

   6. Applied egg-rr 98.9%

      \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{1}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 98.9%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

      2. sin-mult 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]

   8. Applied egg-rr 97.8%

      \[\leadsto 2.6666666666666665 \cdot \frac{1}{\frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]
   9. Step-by-step derivation

      1. div-sub 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]

      2. +-inverses 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      3. cos-0 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      4. metadata-eval 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      5. distribute-lft-out 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}} \]

      6. metadata-eval 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}} \]

      7. *-rgt-identity 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \color{blue}{x}}{2}}} \]

   10. Simplified 97.8%

       \[\leadsto 2.6666666666666665 \cdot \frac{1}{\frac{\sin x}{\color{blue}{0.5 - \frac{\cos x}{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (/ x (* 0.375 (fma (pow x 2.0) -0.3333333333333333 4.0)))
   (* 2.6666666666666665 (/ (- 0.5 (/ (cos x) 2.0)) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = x / (0.375 * fma(pow(x, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) / sin(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(x / Float64(0.375 * fma((x ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / sin(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(x / N[(0.375 * N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 69.3%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. associate-*r/ 69.3%

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

      3. metadata-eval 69.3%

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

      4. clear-num 69.3%

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

      5. *-un-lft-identity 69.3%

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

      6. metadata-eval 69.3%

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

      7. associate-*l* 69.4%

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

      8. times-frac 69.4%

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

      9. metadata-eval 69.4%

         \[\leadsto \frac{1}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

      10. pow2 69.4%

          \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   6. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. Taylor expanded in x around 0 68.3%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 68.3%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 68.3%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 68.3%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{0.375 \cdot \frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. associate-/r* 68.4%

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       3. metadata-eval 68.4%

          \[\leadsto 1 \cdot \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       4. div-inv 68.4%

          \[\leadsto 1 \cdot \color{blue}{\left(2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}\right)} \]

       5. clear-num 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}}\right) \]

       6. +-commutative 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}}\right) \]

       7. fma-define 68.4%

          \[\leadsto 1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}}\right) \]

   11. Applied egg-rr 68.4%

       \[\leadsto \color{blue}{1 \cdot \left(2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\right)} \]
   12. Step-by-step derivation

       1. *-lft-identity 68.4%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       2. metadata-eval 68.4%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       3. times-frac 68.7%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       4. *-lft-identity 68.7%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 68.7%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 0.00419999999999999974 < x

   1. Initial program 98.8%

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

      1. metadata-eval 98.8%

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

      2. associate-*r/ 98.9%

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

      3. associate-*r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r/ 98.8%

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

      6. pow2 98.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x} \cdot 2.6666666666666665} \]
   5. Step-by-step derivation

      1. unpow2 98.9%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}} \]

      2. sin-mult 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]

   6. Applied egg-rr 97.8%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x} \cdot 2.6666666666666665 \]
   7. Step-by-step derivation

      1. div-sub 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right)}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}} \]

      2. +-inverses 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      3. cos-0 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      4. metadata-eval 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{0.5} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      5. distribute-lft-out 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \color{blue}{\left(x \cdot \left(0.5 + 0.5\right)\right)}}{2}}} \]

      6. metadata-eval 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}} \]

      7. *-rgt-identity 98.0%

         \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{0.5 - \frac{\cos \color{blue}{x}}{2}}} \]

   8. Simplified 97.8%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x} \cdot 2.6666666666666665 \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(x \cdot 0.5\right)}{0.75} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.sin((x * 0.5)) / 0.75

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.2%

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

   6. un-div-inv 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. times-frac 99.6%

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

   9. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0 59.0%

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

Alternative 13: 55.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sin (* x 0.5)) 1.3333333333333333))

C

double code(double x) {
	return sin((x * 0.5)) * 1.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin((x * 0.5d0)) * 1.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.sin((x * 0.5)) * 1.3333333333333333;
}

Python

def code(x):
	return math.sin((x * 0.5)) * 1.3333333333333333

Julia

function code(x)
	return Float64(sin(Float64(x * 0.5)) * 1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = sin((x * 0.5)) * 1.3333333333333333;
end

Wolfram

code[x_] := N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 58.8%

   \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{1.3333333333333333} \]
6. Add Preprocessing

Alternative 14: 51.4% 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 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. associate-*r/ 75.4%

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

   3. metadata-eval 75.4%

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

   4. clear-num 75.4%

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

   5. *-un-lft-identity 75.4%

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

   6. metadata-eval 75.4%

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

   7. associate-*l* 75.5%

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

   8. times-frac 75.5%

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

   9. metadata-eval 75.5%

      \[\leadsto \frac{1}{\color{blue}{0.375} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}} \]

   10. pow2 75.5%

       \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

6. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

7. Taylor expanded in x around 0 54.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
8. Add Preprocessing

Alternative 15: 51.3% 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 75.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 54.7%

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

6. Final simplification 54.7%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. Add Preprocessing

Developer Target 1: 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 2024185 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (* 8 (sin (* x 1/2))) 3) (/ (sin x) (sin (* x 1/2)))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))