bug323 (missed optimization)

Percentage Accurate: 6.5% → 10.0%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\\ \pi \cdot 0.5 - {\left(t_0 \cdot t_0\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (sqrt (asin (- 1.0 x))) 0.3333333333333333)))
   (- (* PI 0.5) (pow (* t_0 t_0) 3.0))))

C

double code(double x) {
	double t_0 = pow(sqrt(asin((1.0 - x))), 0.3333333333333333);
	return (((double) M_PI) * 0.5) - pow((t_0 * t_0), 3.0);
}

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.sqrt(Math.asin((1.0 - x))), 0.3333333333333333);
	return (Math.PI * 0.5) - Math.pow((t_0 * t_0), 3.0);
}

Python

def code(x):
	t_0 = math.pow(math.sqrt(math.asin((1.0 - x))), 0.3333333333333333)
	return (math.pi * 0.5) - math.pow((t_0 * t_0), 3.0)

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x))) ^ 0.3333333333333333
	return Float64(Float64(pi * 0.5) - (Float64(t_0 * t_0) ^ 3.0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt(asin((1.0 - x))) ^ 0.3333333333333333;
	tmp = (pi * 0.5) - ((t_0 * t_0) ^ 3.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 6.7%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg 6.7%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

5. Simplified 6.7%

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

   1. add-cube-cbrt 10.2%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow3 10.2%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

7. Applied egg-rr 10.2%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
8. Step-by-step derivation

   1. pow1/3 4.9%

      \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.3333333333333333}\right)}}^{3} \]

   2. add-sqr-sqrt 10.2%

      \[\leadsto \pi \cdot 0.5 - {\left({\color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}}^{0.3333333333333333}\right)}^{3} \]

   3. unpow-prod-down 10.2%

      \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

9. Applied egg-rr 10.2%

   \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

10. Final simplification 10.2%

    \[\leadsto \pi \cdot 0.5 - {\left({\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{3} \]

Alternative 2: 9.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \log \left(\sqrt[3]{e^{t_0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (* PI 0.5) (asin (- 1.0 x)))
     (* 3.0 (log (cbrt (exp t_0)))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = 3.0 * log(cbrt(exp(t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = 3.0 * Math.log(Math.cbrt(Math.exp(t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = Float64(3.0 * log(cbrt(exp(t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(3.0 * N[Log[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. sub-neg 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-cube-cbrt 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. pow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

   7. Applied egg-rr 7.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. add-cube-cbrt 3.9%

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

      3. sub-neg 3.9%

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

      4. add-sqr-sqrt 0.0%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{-\sin^{-1} \left(1 - x\right)} \cdot \sqrt{-\sin^{-1} \left(1 - x\right)}} \]

      5. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\left(-\sin^{-1} \left(1 - x\right)\right) \cdot \left(-\sin^{-1} \left(1 - x\right)\right)}} \]

      6. sqr-neg 6.5%

         \[\leadsto \pi \cdot 0.5 + \sqrt{\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}} \]

      7. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}} \]

      8. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 55.8%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 55.8%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip-- 55.9%

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

      3. div-inv 55.9%

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

      4. metadata-eval 55.9%

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

      5. div-inv 55.9%

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

      6. metadata-eval 55.9%

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

      7. div-inv 55.9%

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

      8. metadata-eval 55.9%

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

   3. Applied egg-rr 55.9%

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

      1. flip-- 55.8%

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

      2. metadata-eval 55.8%

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

      3. div-inv 55.8%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin 55.8%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. add-log-exp 55.8%

         \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]

      6. add-cube-cbrt 56.1%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}} \cdot \sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \cdot \sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

      7. log-prod 56.1%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}} \cdot \sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

      8. pow2 56.1%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \]

   5. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 56.0%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \]

      2. distribute-lft1-in 56.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

      3. metadata-eval 56.0%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right) \]

   7. Simplified 56.0%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\\ \end{array} \]

Alternative 3: 9.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := t_0 + -1\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_1 \cdot t_1}{1 - t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (+ t_0 -1.0)))
   (if (<= t_0 0.0)
     (+ (* PI 0.5) (asin (- 1.0 x)))
     (/ (- 1.0 (* t_1 t_1)) (- 1.0 t_1)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = t_0 + -1.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double t_1 = t_0 + -1.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	t_1 = t_0 + -1.0
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = Float64(t_0 + -1.0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 - t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	t_1 = t_0 + -1.0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = (1.0 - (t_1 * t_1)) / (1.0 - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. sub-neg 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-cube-cbrt 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. pow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

   7. Applied egg-rr 7.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. add-cube-cbrt 3.9%

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

      3. sub-neg 3.9%

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

      4. add-sqr-sqrt 0.0%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{-\sin^{-1} \left(1 - x\right)} \cdot \sqrt{-\sin^{-1} \left(1 - x\right)}} \]

      5. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\left(-\sin^{-1} \left(1 - x\right)\right) \cdot \left(-\sin^{-1} \left(1 - x\right)\right)}} \]

      6. sqr-neg 6.5%

         \[\leadsto \pi \cdot 0.5 + \sqrt{\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}} \]

      7. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}} \]

      8. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 55.8%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 55.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-udef 55.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-udef 55.8%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      4. add-exp-log 55.8%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

   3. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1} \]
   4. Step-by-step derivation

      1. associate--l+ 55.8%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      2. flip-+ 55.9%

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

      3. metadata-eval 55.9%

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

      4. sub-neg 55.9%

         \[\leadsto \frac{1 - \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} \cdot \left(\cos^{-1} \left(1 - x\right) - 1\right)}{1 - \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      5. metadata-eval 55.9%

         \[\leadsto \frac{1 - \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) \cdot \left(\cos^{-1} \left(1 - x\right) - 1\right)}{1 - \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      6. sub-neg 55.9%

         \[\leadsto \frac{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)}}{1 - \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      7. metadata-eval 55.9%

         \[\leadsto \frac{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right)}{1 - \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      8. sub-neg 55.9%

         \[\leadsto \frac{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}{1 - \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)}} \]

      9. metadata-eval 55.9%

         \[\leadsto \frac{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}{1 - \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right)} \]

   5. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right)}\\ \end{array} \]

Alternative 4: 9.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(t_0 + -1\right)}^{2}}{2 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (* PI 0.5) (asin (- 1.0 x)))
     (/ (- 1.0 (pow (+ t_0 -1.0) 2.0)) (- 2.0 t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = (1.0 - pow((t_0 + -1.0), 2.0)) / (2.0 - t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = (1.0 - Math.pow((t_0 + -1.0), 2.0)) / (2.0 - t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = (1.0 - math.pow((t_0 + -1.0), 2.0)) / (2.0 - t_0)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = Float64(Float64(1.0 - (Float64(t_0 + -1.0) ^ 2.0)) / Float64(2.0 - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = (1.0 - ((t_0 + -1.0) ^ 2.0)) / (2.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(t$95$0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. sub-neg 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-cube-cbrt 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. pow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

   7. Applied egg-rr 7.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. add-cube-cbrt 3.9%

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

      3. sub-neg 3.9%

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

      4. add-sqr-sqrt 0.0%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{-\sin^{-1} \left(1 - x\right)} \cdot \sqrt{-\sin^{-1} \left(1 - x\right)}} \]

      5. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\left(-\sin^{-1} \left(1 - x\right)\right) \cdot \left(-\sin^{-1} \left(1 - x\right)\right)}} \]

      6. sqr-neg 6.5%

         \[\leadsto \pi \cdot 0.5 + \sqrt{\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}} \]

      7. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}} \]

      8. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 55.8%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 55.8%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. sub-neg 55.8%

         \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      3. div-inv 55.8%

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

      4. metadata-eval 55.8%

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

   3. Applied egg-rr 55.8%

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

      1. sub-neg 55.8%

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

   5. Simplified 55.8%

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

      1. add-cube-cbrt 56.3%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. pow3 56.3%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

   7. Applied egg-rr 56.3%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow3 56.3%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. add-cube-cbrt 55.8%

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

      3. metadata-eval 55.8%

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

      4. div-inv 55.8%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      5. acos-asin 55.8%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      6. +-rgt-identity 55.8%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + 0} \]

      7. metadata-eval 55.8%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{\left(-1 + 1\right)} \]

      8. associate-+l+ 55.8%

         \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]

      9. flip-+ 55.9%

         \[\leadsto \color{blue}{\frac{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right) - 1 \cdot 1}{\left(\cos^{-1} \left(1 - x\right) + -1\right) - 1}} \]

      10. frac-2neg 55.9%

          \[\leadsto \color{blue}{\frac{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right) - 1 \cdot 1\right)}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) - 1\right)}} \]

      11. metadata-eval 55.9%

          \[\leadsto \frac{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right) - \color{blue}{1}\right)}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) - 1\right)} \]

      12. sub-neg 55.9%

          \[\leadsto \frac{-\color{blue}{\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right) + \left(-1\right)\right)}}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) - 1\right)} \]

      13. pow2 55.9%

          \[\leadsto \frac{-\left(\color{blue}{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}} + \left(-1\right)\right)}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) - 1\right)} \]

      14. metadata-eval 55.9%

          \[\leadsto \frac{-\left({\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2} + \color{blue}{-1}\right)}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) - 1\right)} \]

   9. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{-\left({\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2} + -1\right)}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) + -1\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 55.9%

          \[\leadsto \frac{-\color{blue}{\left(-1 + {\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}\right)}}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) + -1\right)} \]

       2. distribute-neg-in 55.9%

          \[\leadsto \frac{\color{blue}{\left(--1\right) + \left(-{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}\right)}}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) + -1\right)} \]

       3. metadata-eval 55.9%

          \[\leadsto \frac{\color{blue}{1} + \left(-{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}\right)}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) + -1\right)} \]

       4. sub-neg 55.9%

          \[\leadsto \frac{\color{blue}{1 - {\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}}{-\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) + -1\right)} \]

       5. neg-sub0 55.9%

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

       6. +-commutative 55.9%

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

       7. associate--r+ 55.9%

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

       8. metadata-eval 55.9%

          \[\leadsto \frac{1 - {\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}{\color{blue}{1} - \left(\cos^{-1} \left(1 - x\right) + -1\right)} \]

       9. +-commutative 55.9%

          \[\leadsto \frac{1 - {\color{blue}{\left(-1 + \cos^{-1} \left(1 - x\right)\right)}}^{2}}{1 - \left(\cos^{-1} \left(1 - x\right) + -1\right)} \]

       10. +-commutative 55.9%

           \[\leadsto \frac{1 - {\left(-1 + \cos^{-1} \left(1 - x\right)\right)}^{2}}{1 - \color{blue}{\left(-1 + \cos^{-1} \left(1 - x\right)\right)}} \]

       11. associate--r+ 55.9%

           \[\leadsto \frac{1 - {\left(-1 + \cos^{-1} \left(1 - x\right)\right)}^{2}}{\color{blue}{\left(1 - -1\right) - \cos^{-1} \left(1 - x\right)}} \]

       12. metadata-eval 55.9%

           \[\leadsto \frac{1 - {\left(-1 + \cos^{-1} \left(1 - x\right)\right)}^{2}}{\color{blue}{2} - \cos^{-1} \left(1 - x\right)} \]

   11. Simplified 55.9%

       \[\leadsto \color{blue}{\frac{1 - {\left(-1 + \cos^{-1} \left(1 - x\right)\right)}^{2}}{2 - \cos^{-1} \left(1 - x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}{2 - \cos^{-1} \left(1 - x\right)}\\ \end{array} \]

Alternative 5: 9.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \log \left(e^{t_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (+ (* PI 0.5) (asin (- 1.0 x)))
     (+ -1.0 (+ 1.0 (log (exp t_0)))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = -1.0 + (1.0 + log(exp(t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = -1.0 + (1.0 + Math.log(Math.exp(t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = -1.0 + (1.0 + math.log(math.exp(t_0)))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = Float64(-1.0 + Float64(1.0 + log(exp(t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = -1.0 + (1.0 + log(exp(t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(1.0 + N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. sub-neg 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-cube-cbrt 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. pow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

   7. Applied egg-rr 7.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. add-cube-cbrt 3.9%

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

      3. sub-neg 3.9%

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

      4. add-sqr-sqrt 0.0%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{-\sin^{-1} \left(1 - x\right)} \cdot \sqrt{-\sin^{-1} \left(1 - x\right)}} \]

      5. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\left(-\sin^{-1} \left(1 - x\right)\right) \cdot \left(-\sin^{-1} \left(1 - x\right)\right)}} \]

      6. sqr-neg 6.5%

         \[\leadsto \pi \cdot 0.5 + \sqrt{\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}} \]

      7. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}} \]

      8. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 55.8%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 55.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-udef 55.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-udef 55.8%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      4. add-exp-log 55.8%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

   3. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1} \]
   4. Step-by-step derivation

      1. add-log-exp 55.9%

         \[\leadsto \left(1 + \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)}\right) - 1 \]

   5. Applied egg-rr 55.9%

      \[\leadsto \left(1 + \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)\\ \end{array} \]

Alternative 6: 10.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 6.7%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg 6.7%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. sub-neg 6.7%

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

5. Simplified 6.7%

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

   1. add-cube-cbrt 10.2%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow3 10.2%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

7. Applied egg-rr 10.2%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

8. Final simplification 10.2%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 7: 9.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (+ (* PI 0.5) (asin (- 1.0 x))) (+ -1.0 (+ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = -1.0 + (1.0 + t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = -1.0 + (1.0 + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = -1.0 + (1.0 + t_0)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = Float64(-1.0 + Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = -1.0 + (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. sub-neg 3.9%

         \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

      1. add-cube-cbrt 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. pow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

   7. Applied egg-rr 7.5%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow3 7.5%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

      2. add-cube-cbrt 3.9%

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

      3. sub-neg 3.9%

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

      4. add-sqr-sqrt 0.0%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{-\sin^{-1} \left(1 - x\right)} \cdot \sqrt{-\sin^{-1} \left(1 - x\right)}} \]

      5. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\left(-\sin^{-1} \left(1 - x\right)\right) \cdot \left(-\sin^{-1} \left(1 - x\right)\right)}} \]

      6. sqr-neg 6.5%

         \[\leadsto \pi \cdot 0.5 + \sqrt{\color{blue}{\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}} \]

      7. sqrt-unprod 6.5%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}} \]

      8. add-sqr-sqrt 6.5%

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

   9. Applied egg-rr 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 55.8%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 55.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-udef 55.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-udef 55.8%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      4. add-exp-log 55.8%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

   3. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)\\ \end{array} \]

Alternative 8: 6.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (+ (acos (- 1.0 x)) -1.0)))

C

double code(double x) {
	return 1.0 + (acos((1.0 - x)) + -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (acos((1.0d0 - x)) + (-1.0d0))
end function

Java

public static double code(double x) {
	return 1.0 + (Math.acos((1.0 - x)) + -1.0);
}

Python

def code(x):
	return 1.0 + (math.acos((1.0 - x)) + -1.0)

Julia

function code(x)
	return Float64(1.0 + Float64(acos(Float64(1.0 - x)) + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (acos((1.0 - x)) + -1.0);
end

Wolfram

code[x_] := N[(1.0 + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 6.7%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. flip-- 6.7%

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

   3. div-inv 6.7%

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

   4. metadata-eval 6.7%

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

   5. div-inv 6.7%

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

   6. metadata-eval 6.7%

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

   7. div-inv 6.7%

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

   8. metadata-eval 6.7%

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

3. Applied egg-rr 6.7%

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

   1. flip-- 6.7%

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

   2. metadata-eval 6.7%

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

   3. div-inv 6.7%

      \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

   4. acos-asin 6.7%

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   5. expm1-log1p-u 6.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   6. expm1-udef 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   7. log1p-udef 6.7%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

   8. add-exp-log 6.7%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

   9. associate--l+ 6.7%

      \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

   10. +-commutative 6.7%

       \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

   11. sub-neg 6.7%

       \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

   12. metadata-eval 6.7%

       \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

5. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]

6. Final simplification 6.7%

   \[\leadsto 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \]

Alternative 9: 6.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 + \cos^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (+ 1.0 (acos (- 1.0 x)))))

C

double code(double x) {
	return -1.0 + (1.0 + acos((1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (1.0d0 + acos((1.0d0 - x)))
end function

Java

public static double code(double x) {
	return -1.0 + (1.0 + Math.acos((1.0 - x)));
}

Python

def code(x):
	return -1.0 + (1.0 + math.acos((1.0 - x)))

Julia

function code(x)
	return Float64(-1.0 + Float64(1.0 + acos(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (1.0 + acos((1.0 - x)));
end

Wolfram

code[x_] := N[(-1.0 + N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. expm1-log1p-u 6.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   2. expm1-udef 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   3. log1p-udef 6.7%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

   4. add-exp-log 6.7%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

3. Applied egg-rr 6.7%

   \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1} \]

4. Final simplification 6.7%

   \[\leadsto -1 + \left(1 + \cos^{-1} \left(1 - x\right)\right) \]

Alternative 10: 6.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]

2. Final simplification 6.7%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]

Developer target: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (asin (sqrt (/ x 2.0)))))

C

double code(double x) {
	return 2.0 * asin(sqrt((x / 2.0)));
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0 * math.asin(math.sqrt((x / 2.0)))

Julia

function code(x)
	return Float64(2.0 * asin(sqrt(Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * asin(sqrt((x / 2.0)));
end

Wolfram

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

Reproduce

herbie shell --seed 2023174 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))