bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 17.4s

Alternatives: 5

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: 6.9% 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: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow (pow (* PI 0.5) 0.3333333333333333) 2.0)
  (cbrt (* PI 0.5))
  (- (asin (- 1.0 x)))))

C

double code(double x) {
	return fma(pow(pow((((double) M_PI) * 0.5), 0.3333333333333333), 2.0), cbrt((((double) M_PI) * 0.5)), -asin((1.0 - x)));
}

Julia

function code(x)
	return fma(((Float64(pi * 0.5) ^ 0.3333333333333333) ^ 2.0), cbrt(Float64(pi * 0.5)), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. acos-asin 8.3%

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

   2. add-cube-cbrt 6.5%

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

   3. fma-neg 6.5%

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

   4. pow2 6.5%

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

   5. div-inv 6.5%

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

   6. metadata-eval 6.5%

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

   7. div-inv 6.5%

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

   8. metadata-eval 6.5%

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

3. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right)} \]
4. Step-by-step derivation

   1. pow1/3 11.9%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

5. Applied egg-rr 11.9%

   \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

6. Final simplification 11.9%

   \[\leadsto \mathsf{fma}\left({\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

Alternative 2: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. rem-exp-log 8.3%

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

3. Applied egg-rr 8.3%

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

   1. add-exp-log 8.3%

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

   2. log1p-udef 8.3%

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

   3. expm1-udef 8.3%

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

   4. expm1-log1p-u 8.3%

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

   5. acos-asin 8.3%

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

   6. div-inv 8.3%

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

   7. metadata-eval 8.3%

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

   8. add-cbrt-cube 6.5%

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

   9. unpow2 6.5%

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

   10. cbrt-prod 11.8%

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

   11. cancel-sign-sub-inv 11.8%

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

   12. unpow2 11.8%

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

   13. cbrt-prod 11.8%

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

   14. pow2 11.8%

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

5. Applied egg-rr 11.8%

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

6. Final simplification 11.8%

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

Alternative 3: 9.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \cos^{-1} \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (+ (* PI 0.5) (asin (- 1.0 x)))
   (exp (log (acos (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = exp(log(acos((1.0 - x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = Math.exp(Math.log(Math.acos((1.0 - x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = math.exp(math.log(math.acos((1.0 - x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = exp(log(acos(Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = exp(log(acos((1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. expm1-log1p-u 3.8%

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

      2. expm1-udef 3.8%

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

      3. log1p-udef 3.8%

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

      4. rem-exp-log 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. add-exp-log 3.8%

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

      2. log1p-udef 3.8%

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

      3. expm1-udef 3.8%

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

      4. expm1-log1p-u 3.8%

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

      5. acos-asin 3.8%

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

      6. div-inv 3.8%

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

      7. metadata-eval 3.8%

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

      8. add-sqr-sqrt 7.6%

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

      9. cancel-sign-sub-inv 7.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 6.7%

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

      12. sqr-neg 6.7%

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

      13. add-sqr-sqrt 6.7%

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

      14. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 81.1%

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

      1. add-exp-log 81.1%

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

   3. Applied egg-rr 81.1%

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

4. Final simplification 11.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \cos^{-1} \left(1 - x\right)}\\ \end{array} \]

Alternative 4: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (+ (* PI 0.5) (asin (- 1.0 x))) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. expm1-log1p-u 3.8%

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

      2. expm1-udef 3.8%

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

      3. log1p-udef 3.8%

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

      4. rem-exp-log 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. add-exp-log 3.8%

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

      2. log1p-udef 3.8%

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

      3. expm1-udef 3.8%

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

      4. expm1-log1p-u 3.8%

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

      5. acos-asin 3.8%

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

      6. div-inv 3.8%

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

      7. metadata-eval 3.8%

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

      8. add-sqr-sqrt 7.6%

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

      9. cancel-sign-sub-inv 7.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 6.7%

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

      12. sqr-neg 6.7%

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

      13. add-sqr-sqrt 6.7%

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

      14. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 81.1%

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

4. Final simplification 11.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 5: 6.9% 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 8.3%

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

2. Final simplification 8.3%

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

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 2023315 
(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)))