bug323 (missed optimization)

Percentage Accurate: 7.3% → 10.9%

Time: 14.0s

Alternatives: 11

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: 7.3% 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: 10.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{t_0}^{2}\right)}{\pi \cdot 0.5 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (fma
     (cbrt (pow (* PI 0.5) 4.0))
     (cbrt (* 0.25 (pow PI 2.0)))
     (- (pow t_0 2.0)))
    (+ (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(cbrt(pow((((double) M_PI) * 0.5), 4.0)), cbrt((0.25 * pow(((double) M_PI), 2.0))), -pow(t_0, 2.0)) / ((((double) M_PI) * 0.5) + t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(cbrt((Float64(pi * 0.5) ^ 4.0)), cbrt(Float64(0.25 * (pi ^ 2.0))), Float64(-(t_0 ^ 2.0))) / Float64(Float64(pi * 0.5) + t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + (-N[Power[t$95$0, 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. acos-asin 7.0%

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

   2. flip-- 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

   5. div-inv 7.0%

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

   6. metadata-eval 7.0%

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

   7. div-inv 7.0%

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

   8. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

5. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 7.0%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}} - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. fma-neg 10.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, -{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr 10.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

8. Final simplification 10.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

Alternative 2: 10.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t_0, {t_0}^{2}, {t_0}^{3}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))))
   (+ (acos (- 1.0 x)) (fma (- t_0) (pow t_0 2.0) (pow t_0 3.0)))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return acos((1.0 - x)) + fma(-t_0, pow(t_0, 2.0), pow(t_0, 3.0));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), (t_0 ^ 2.0), (t_0 ^ 3.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$0) * N[Power[t$95$0, 2.0], $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. expm1-log1p-u 7.0%

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

   2. expm1-udef 7.0%

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

   3. log1p-udef 7.0%

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

   4. add-exp-log 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-exp-log 7.0%

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

   2. log1p-udef 7.0%

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

   3. expm1-udef 7.0%

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

   4. expm1-log1p-u 7.0%

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

   5. acos-asin 7.0%

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

   6. div-inv 7.0%

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

   7. metadata-eval 7.0%

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

   8. add-cube-cbrt 10.2%

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

   9. prod-diff 10.2%

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

5. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 10.2%

      \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right) \]

   2. pow3 10.2%

      \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right) \]

7. Applied egg-rr 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right) \]

8. Final simplification 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}\right) \]

Alternative 3: 10.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - t_0 \cdot {\left(\sqrt{t_0}\right)}^{2}}{\pi \cdot 0.5 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (* PI 0.5) (* PI 0.5)) (* t_0 (pow (sqrt t_0) 2.0)))
    (+ (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (((((double) M_PI) * 0.5) * (((double) M_PI) * 0.5)) - (t_0 * pow(sqrt(t_0), 2.0))) / ((((double) M_PI) * 0.5) + t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	return (((Math.PI * 0.5) * (Math.PI * 0.5)) - (t_0 * Math.pow(Math.sqrt(t_0), 2.0))) / ((Math.PI * 0.5) + t_0);
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	return (((math.pi * 0.5) * (math.pi * 0.5)) - (t_0 * math.pow(math.sqrt(t_0), 2.0))) / ((math.pi * 0.5) + t_0)

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64(pi * 0.5) * Float64(pi * 0.5)) - Float64(t_0 * (sqrt(t_0) ^ 2.0))) / Float64(Float64(pi * 0.5) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = asin((1.0 - x));
	tmp = (((pi * 0.5) * (pi * 0.5)) - (t_0 * (sqrt(t_0) ^ 2.0))) / ((pi * 0.5) + t_0);
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. acos-asin 7.0%

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

   2. flip-- 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

   5. div-inv 7.0%

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

   6. metadata-eval 7.0%

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

   7. div-inv 7.0%

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

   8. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

5. Applied egg-rr 10.2%

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

6. Final simplification 10.2%

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

Alternative 4: 10.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\sqrt[3]{0.25} \cdot \left({\pi}^{2} \cdot \sqrt[3]{0.0625}\right) - {t_0}^{2}}{\pi \cdot 0.5 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (cbrt 0.25) (* (pow PI 2.0) (cbrt 0.0625))) (pow t_0 2.0))
    (+ (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return ((cbrt(0.25) * (pow(((double) M_PI), 2.0) * cbrt(0.0625))) - pow(t_0, 2.0)) / ((((double) M_PI) * 0.5) + t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	return ((Math.cbrt(0.25) * (Math.pow(Math.PI, 2.0) * Math.cbrt(0.0625))) - Math.pow(t_0, 2.0)) / ((Math.PI * 0.5) + t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(cbrt(0.25) * Float64((pi ^ 2.0) * cbrt(0.0625))) - (t_0 ^ 2.0)) / Float64(Float64(pi * 0.5) + t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Power[0.25, 1/3], $MachinePrecision] * N[(N[Power[Pi, 2.0], $MachinePrecision] * N[Power[0.0625, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. acos-asin 7.0%

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

   2. flip-- 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

   5. div-inv 7.0%

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

   6. metadata-eval 7.0%

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

   7. div-inv 7.0%

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

   8. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-sqr-sqrt 10.2%

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

   2. pow2 10.2%

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

5. Applied egg-rr 10.2%

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

   1. add-cube-cbrt 7.0%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}} - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. fma-neg 10.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, -{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr 10.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

8. Taylor expanded in x around 0 7.0%

   \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot \left({\pi}^{2} \cdot \left(\sqrt[3]{0.0625} \cdot \sqrt[3]{0.25}\right)\right) - {\sin^{-1} \left(1 - x\right)}^{2}}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
9. Step-by-step derivation

   1. pow-base-1 7.0%

      \[\leadsto \frac{\color{blue}{1} \cdot \left({\pi}^{2} \cdot \left(\sqrt[3]{0.0625} \cdot \sqrt[3]{0.25}\right)\right) - {\sin^{-1} \left(1 - x\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. *-lft-identity 7.0%

      \[\leadsto \frac{\color{blue}{{\pi}^{2} \cdot \left(\sqrt[3]{0.0625} \cdot \sqrt[3]{0.25}\right)} - {\sin^{-1} \left(1 - x\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   3. associate-*r* 10.2%

      \[\leadsto \frac{\color{blue}{\left({\pi}^{2} \cdot \sqrt[3]{0.0625}\right) \cdot \sqrt[3]{0.25}} - {\sin^{-1} \left(1 - x\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   4. *-commutative 10.2%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{0.25} \cdot \left({\pi}^{2} \cdot \sqrt[3]{0.0625}\right)} - {\sin^{-1} \left(1 - x\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

10. Simplified 10.2%

    \[\leadsto \frac{\color{blue}{\sqrt[3]{0.25} \cdot \left({\pi}^{2} \cdot \sqrt[3]{0.0625}\right) - {\sin^{-1} \left(1 - x\right)}^{2}}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

11. Final simplification 10.2%

    \[\leadsto \frac{\sqrt[3]{0.25} \cdot \left({\pi}^{2} \cdot \sqrt[3]{0.0625}\right) - {\sin^{-1} \left(1 - x\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

Alternative 5: 10.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t_1, t_1, t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (acos (- 1.0 x)) (fma (- t_1) t_1 t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return acos((1.0 - x)) + fma(-t_1, t_1, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), t_1, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.0%

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

   1. expm1-log1p-u 7.0%

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

   2. expm1-udef 7.0%

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

   3. log1p-udef 7.0%

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

   4. add-exp-log 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-exp-log 7.0%

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

   2. log1p-udef 7.0%

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

   3. expm1-udef 7.0%

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

   4. expm1-log1p-u 7.0%

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

   5. acos-asin 7.0%

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

   6. div-inv 7.0%

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

   7. metadata-eval 7.0%

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

   8. add-sqr-sqrt 10.2%

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

   9. prod-diff 10.2%

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

   10. add-sqr-sqrt 10.2%

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

   11. fma-neg 10.2%

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

   12. metadata-eval 10.2%

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

   13. div-inv 10.2%

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

   14. acos-asin 10.3%

       \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \]

5. Applied egg-rr 10.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right)} \]

6. Final simplification 10.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

Alternative 6: 7.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\frac{1}{\frac{2 + t_1}{{\left(1 + t_1\right)}^{2} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(\pi \cdot 0.5, t_0\right)\right)}^{2}}{\pi \cdot 0.5 + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0)
     (/ 1.0 (/ (+ 2.0 t_1) (+ (pow (+ 1.0 t_1) 2.0) -1.0)))
     (/ (pow (hypot (* PI 0.5) t_0) 2.0) (+ (* PI 0.5) t_0)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 / ((2.0 + t_1) / (pow((1.0 + t_1), 2.0) + -1.0));
	} else {
		tmp = pow(hypot((((double) M_PI) * 0.5), t_0), 2.0) / ((((double) M_PI) * 0.5) + t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double t_1 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 / ((2.0 + t_1) / (Math.pow((1.0 + t_1), 2.0) + -1.0));
	} else {
		tmp = Math.pow(Math.hypot((Math.PI * 0.5), t_0), 2.0) / ((Math.PI * 0.5) + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	t_1 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = 1.0 / ((2.0 + t_1) / (math.pow((1.0 + t_1), 2.0) + -1.0))
	else:
		tmp = math.pow(math.hypot((math.pi * 0.5), t_0), 2.0) / ((math.pi * 0.5) + t_0)
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(1.0 / Float64(Float64(2.0 + t_1) / Float64((Float64(1.0 + t_1) ^ 2.0) + -1.0)));
	else
		tmp = Float64((hypot(Float64(pi * 0.5), t_0) ^ 2.0) / Float64(Float64(pi * 0.5) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	t_1 = acos((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = 1.0 / ((2.0 + t_1) / (((1.0 + t_1) ^ 2.0) + -1.0));
	else
		tmp = (hypot((pi * 0.5), t_0) ^ 2.0) / ((pi * 0.5) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(1.0 / N[(N[(2.0 + t$95$1), $MachinePrecision] / N[(N[Power[N[(1.0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.0%

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

      1. expm1-log1p-u 7.0%

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

      2. expm1-udef 7.0%

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

      3. log1p-udef 7.0%

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

      4. add-exp-log 7.0%

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

   3. Applied egg-rr 7.0%

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

      1. flip-- 7.0%

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

      2. clear-num 7.0%

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

      3. +-commutative 7.0%

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

      4. associate-+l+ 7.0%

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

      5. metadata-eval 7.0%

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

      6. metadata-eval 7.0%

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

      7. sub-neg 7.0%

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

      8. pow2 7.0%

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

      9. metadata-eval 7.0%

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

   5. Applied egg-rr 7.0%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.0%

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

      1. acos-asin 7.0%

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

      2. flip-- 7.0%

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

      3. div-inv 7.0%

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

      4. metadata-eval 7.0%

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

      5. div-inv 7.0%

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

      6. metadata-eval 7.0%

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

      7. div-inv 7.0%

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

      8. metadata-eval 7.0%

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

   3. Applied egg-rr 7.0%

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

      1. add-sqr-sqrt 10.2%

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

      2. pow2 10.2%

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

   5. Applied egg-rr 10.2%

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

      1. add-cube-cbrt 7.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}} - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

      2. fma-neg 10.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)} \cdot \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, \sqrt[3]{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)}, -{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   7. Applied egg-rr 10.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{4}}, \sqrt[3]{0.25 \cdot {\pi}^{2}}, -{\sin^{-1} \left(1 - x\right)}^{2}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   9. Applied egg-rr 6.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}\right)} - 1}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
   10. Step-by-step derivation

       1. expm1-def 6.8%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

       2. expm1-log1p 6.8%

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

   11. Simplified 6.8%

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\right)}^{2}}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.0%

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

Alternative 7: 9.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2 + t_0}{{\left(1 + t_0\right)}^{2} + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17)
     (+ t_0 (* 2.0 (asin (- 1.0 x))))
     (/ 1.0 (/ (+ 2.0 t_0) (+ (pow (+ 1.0 t_0) 2.0) -1.0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = 1.0 / ((2.0 + t_0) / (pow((1.0 + t_0), 2.0) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.5d-17) then
        tmp = t_0 + (2.0d0 * asin((1.0d0 - x)))
    else
        tmp = 1.0d0 / ((2.0d0 + t_0) / (((1.0d0 + t_0) ** 2.0d0) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (2.0 * Math.asin((1.0 - x)));
	} else {
		tmp = 1.0 / ((2.0 + t_0) / (Math.pow((1.0 + t_0), 2.0) + -1.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = t_0 + (2.0 * math.asin((1.0 - x)))
	else:
		tmp = 1.0 / ((2.0 + t_0) / (math.pow((1.0 + t_0), 2.0) + -1.0))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(2.0 * asin(Float64(1.0 - x))));
	else
		tmp = Float64(1.0 / Float64(Float64(2.0 + t_0) / Float64((Float64(1.0 + t_0) ^ 2.0) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	else
		tmp = 1.0 / ((2.0 + t_0) / (((1.0 + t_0) ^ 2.0) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(2.0 + t$95$0), $MachinePrecision] / N[(N[Power[N[(1.0 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.2%

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

      9. prod-diff 7.2%

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

   5. Applied egg-rr 7.2%

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

      1. fma-udef 7.2%

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

   7. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. cube-mult 6.4%

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

      3. rem-cbrt-cube 6.4%

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

      4. count-2 6.4%

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

   9. Simplified 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

      1. expm1-log1p-u 60.8%

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

      2. expm1-udef 61.0%

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

      3. log1p-udef 61.0%

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

      4. add-exp-log 61.0%

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

   3. Applied egg-rr 61.0%

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

      1. flip-- 60.9%

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

      2. clear-num 60.9%

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

      3. +-commutative 60.9%

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

      4. associate-+l+ 61.0%

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

      5. metadata-eval 61.0%

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

      6. metadata-eval 61.0%

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

      7. sub-neg 61.0%

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

      8. pow2 61.0%

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

      9. metadata-eval 61.0%

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

   5. Applied egg-rr 61.0%

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

4. Final simplification 9.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2 + \cos^{-1} \left(1 - x\right)}{{\left(1 + \cos^{-1} \left(1 - x\right)\right)}^{2} + -1}}\\ \end{array} \]

Alternative 8: 9.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \log \left(e^{t_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17)
     (+ t_0 (* 2.0 (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 (x <= 5.5e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = 1.0 + (-1.0 + log(exp(t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.5d-17) then
        tmp = t_0 + (2.0d0 * asin((1.0d0 - x)))
    else
        tmp = 1.0d0 + ((-1.0d0) + log(exp(t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (2.0 * 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 x <= 5.5e-17:
		tmp = t_0 + (2.0 * 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 (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(2.0 * 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 (x <= 5.5e-17)
		tmp = t_0 + (2.0 * 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[x, 5.5e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $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 x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.2%

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

      9. prod-diff 7.2%

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

   5. Applied egg-rr 7.2%

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

      1. fma-udef 7.2%

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

   7. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. cube-mult 6.4%

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

      3. rem-cbrt-cube 6.4%

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

      4. count-2 6.4%

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

   9. Simplified 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

      1. expm1-log1p-u 60.8%

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

      2. expm1-udef 61.0%

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

      3. log1p-udef 61.0%

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

      4. add-exp-log 61.0%

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

   3. Applied egg-rr 61.0%

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

      1. associate--l+ 60.8%

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

      2. +-commutative 60.8%

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

      3. sub-neg 60.8%

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

      4. metadata-eval 60.8%

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

   5. Applied egg-rr 60.8%

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

      1. add-log-exp 61.0%

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

   7. Applied egg-rr 61.0%

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

4. Final simplification 9.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\right)\\ \end{array} \]

Alternative 9: 9.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t_0 + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17) (+ t_0 (* 2.0 (asin (- 1.0 x)))) (+ (+ 1.0 t_0) -1.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	} else {
		tmp = (1.0 + t_0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.5d-17) then
        tmp = t_0 + (2.0d0 * asin((1.0d0 - x)))
    else
        tmp = (1.0d0 + t_0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = t_0 + (2.0 * Math.asin((1.0 - x)));
	} else {
		tmp = (1.0 + t_0) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = t_0 + (2.0 * math.asin((1.0 - x)))
	else:
		tmp = (1.0 + t_0) + -1.0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(t_0 + Float64(2.0 * asin(Float64(1.0 - x))));
	else
		tmp = Float64(Float64(1.0 + t_0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = t_0 + (2.0 * asin((1.0 - x)));
	else
		tmp = (1.0 + t_0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(t$95$0 + N[(2.0 * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.9%

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

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-exp-log 3.9%

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

      2. log1p-udef 3.9%

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

      3. expm1-udef 3.9%

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

      4. expm1-log1p-u 3.9%

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

      5. acos-asin 3.9%

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

      6. div-inv 3.9%

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

      7. metadata-eval 3.9%

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

      8. add-cube-cbrt 7.2%

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

      9. prod-diff 7.2%

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

   5. Applied egg-rr 7.2%

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

      1. fma-udef 7.2%

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

   7. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. cube-mult 6.4%

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

      3. rem-cbrt-cube 6.4%

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

      4. count-2 6.4%

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

   9. Simplified 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.8%

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

      1. expm1-log1p-u 60.8%

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

      2. expm1-udef 61.0%

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

      3. log1p-udef 61.0%

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

      4. add-exp-log 61.0%

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

   3. Applied egg-rr 61.0%

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

4. Final simplification 9.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right) + 2 \cdot \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\\ \end{array} \]

Alternative 10: 7.3% accurate, 1.0× speedup

Math

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

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(Float64(1.0 + 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[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. expm1-log1p-u 7.0%

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

   2. expm1-udef 7.0%

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

   3. log1p-udef 7.0%

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

   4. add-exp-log 7.0%

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

3. Applied egg-rr 7.0%

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

4. Final simplification 7.0%

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

Alternative 11: 7.3% 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 7.0%

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

2. Final simplification 7.0%

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

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 2023278 
(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)))