bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 8.5s

Alternatives: 8

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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ t_1 := {t\_0}^{2}\\ \mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, \sqrt[3]{\pi \cdot 0.5}, -t\_0 \cdot t\_1\right) + \mathsf{fma}\left(-t\_0, t\_1, t\_1 \cdot \sqrt[3]{\sin^{-1} \log \left(e^{1 - x}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x)))) (t_1 (pow t_0 2.0)))
   (+
    (fma (cbrt (pow (* PI 0.5) 2.0)) (cbrt (* PI 0.5)) (- (* t_0 t_1)))
    (fma (- t_0) t_1 (* t_1 (cbrt (asin (log (exp (- 1.0 x))))))))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	double t_1 = pow(t_0, 2.0);
	return fma(cbrt(pow((((double) M_PI) * 0.5), 2.0)), cbrt((((double) M_PI) * 0.5)), -(t_0 * t_1)) + fma(-t_0, t_1, (t_1 * cbrt(asin(log(exp((1.0 - x)))))));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	t_1 = t_0 ^ 2.0
	return Float64(fma(cbrt((Float64(pi * 0.5) ^ 2.0)), cbrt(Float64(pi * 0.5)), Float64(-Float64(t_0 * t_1))) + fma(Float64(-t_0), t_1, Float64(t_1 * cbrt(asin(log(exp(Float64(1.0 - x))))))))
end

Wolfram

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

Derivation

1. Initial program 5.8%

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

   1. acos-asin 5.8%

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

   2. add-cube-cbrt 3.9%

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

   3. add-cube-cbrt 5.8%

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

   4. prod-diff 5.8%

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

4. Applied egg-rr 9.7%

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

   1. add-log-exp 9.7%

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

6. Applied egg-rr 9.7%

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

7. Final simplification 9.7%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, \sqrt[3]{\pi \cdot 0.5}, -\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(1 - x\right)}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}, {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sqrt[3]{\sin^{-1} \log \left(e^{1 - x}\right)}\right) \]
8. Add Preprocessing

Alternative 2: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{\mathsf{fma}\left(t\_0, t\_0, -\sin^{-1} \left(1 - x\right)\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5))))
   (+
    (log (cbrt (pow (exp (acos (- 1.0 x))) 2.0)))
    (log (cbrt (exp (fma t_0 t_0 (- (asin (- 1.0 x))))))))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	return log(cbrt(pow(exp(acos((1.0 - x))), 2.0))) + log(cbrt(exp(fma(t_0, t_0, -asin((1.0 - x))))));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	return Float64(log(cbrt((exp(acos(Float64(1.0 - x))) ^ 2.0))) + log(cbrt(exp(fma(t_0, t_0, Float64(-asin(Float64(1.0 - x))))))))
end

Wolfram

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

Derivation

1. Initial program 5.8%

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

   1. add-log-exp 5.8%

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

   2. add-cube-cbrt 5.8%

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

   3. log-prod 5.8%

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

   4. cbrt-unprod 5.8%

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

   5. pow2 5.8%

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

4. Applied egg-rr 5.8%

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

   1. acos-asin 5.8%

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

   2. add-sqr-sqrt 9.7%

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

   3. fma-neg 9.7%

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

   4. div-inv 9.7%

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

   5. metadata-eval 9.7%

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

   6. div-inv 9.7%

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

   7. metadata-eval 9.7%

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

6. Applied egg-rr 9.7%

   \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\mathsf{fma}\left(\sqrt{\pi \cdot 0.5}, \sqrt{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right)}}}\right) \]
7. Add Preprocessing

Alternative 3: 9.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos x) (log (exp (acos (- 1.0 x))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $MachinePrecision], N[Log[N[Exp[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. Add Preprocessing

   3. Taylor expanded in x around inf 6.7%

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

      1. neg-mul-1 6.7%

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

   5. Simplified 6.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.7%

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

      3. sqr-neg 6.7%

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

      4. sqrt-unprod 6.7%

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

      5. add-sqr-sqrt 6.7%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.7%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.7%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.7%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 74.9%

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

      1. add-log-exp 75.2%

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

   4. Applied egg-rr 75.2%

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

Alternative 4: 9.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos x) (exp (log (acos (- 1.0 x))))))

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(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.acos(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 = acos(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 = acos(x);
	else
		tmp = exp(log(acos((1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $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. Add Preprocessing

   3. Taylor expanded in x around inf 6.7%

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

      1. neg-mul-1 6.7%

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

   5. Simplified 6.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.7%

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

      3. sqr-neg 6.7%

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

      4. sqrt-unprod 6.7%

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

      5. add-sqr-sqrt 6.7%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.7%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.7%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.7%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 74.9%

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

      1. add-exp-log 74.9%

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

   4. Applied egg-rr 74.9%

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

Alternative 5: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\pi}^{3} \cdot 0.125} - \sin^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (cbrt (* (pow PI 3.0) 0.125)) (asin (- 1.0 x))))

C

double code(double x) {
	return cbrt((pow(((double) M_PI), 3.0) * 0.125)) - asin((1.0 - x));
}

Java

public static double code(double x) {
	return Math.cbrt((Math.pow(Math.PI, 3.0) * 0.125)) - Math.asin((1.0 - x));
}

Julia

function code(x)
	return Float64(cbrt(Float64((pi ^ 3.0) * 0.125)) - asin(Float64(1.0 - x)))
end

Wolfram

code[x_] := N[(N[Power[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision], 1/3], $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

   1. acos-asin 5.8%

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

   2. add-cube-cbrt 3.9%

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

   3. add-cube-cbrt 5.8%

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

   4. prod-diff 5.8%

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

4. Applied egg-rr 9.7%

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

5. Taylor expanded in x around 0 5.8%

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

7. Simplified 9.7%

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

   1. add-cbrt-cube 9.7%

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

   2. pow1/3 9.7%

      \[\leadsto \color{blue}{{\left(\left(\left(\left(\pi \cdot \sqrt[3]{0.25}\right) \cdot \sqrt[3]{0.5}\right) \cdot \left(\left(\pi \cdot \sqrt[3]{0.25}\right) \cdot \sqrt[3]{0.5}\right)\right) \cdot \left(\left(\pi \cdot \sqrt[3]{0.25}\right) \cdot \sqrt[3]{0.5}\right)\right)}^{0.3333333333333333}} - \sin^{-1} \left(1 - x\right) \]

   3. pow3 9.7%

      \[\leadsto {\color{blue}{\left({\left(\left(\pi \cdot \sqrt[3]{0.25}\right) \cdot \sqrt[3]{0.5}\right)}^{3}\right)}}^{0.3333333333333333} - \sin^{-1} \left(1 - x\right) \]

   4. associate-*l* 5.8%

      \[\leadsto {\left({\color{blue}{\left(\pi \cdot \left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.5}\right)\right)}}^{3}\right)}^{0.3333333333333333} - \sin^{-1} \left(1 - x\right) \]

   5. unpow-prod-down 5.8%

      \[\leadsto {\color{blue}{\left({\pi}^{3} \cdot {\left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.5}\right)}^{3}\right)}}^{0.3333333333333333} - \sin^{-1} \left(1 - x\right) \]

   6. metadata-eval 5.8%

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

   7. cbrt-unprod 9.7%

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

   8. rem-3cbrt-lft 5.8%

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

   9. metadata-eval 5.8%

      \[\leadsto {\left({\pi}^{3} \cdot \color{blue}{0.125}\right)}^{0.3333333333333333} - \sin^{-1} \left(1 - x\right) \]

9. Applied egg-rr 5.8%

   \[\leadsto \color{blue}{{\left({\pi}^{3} \cdot 0.125\right)}^{0.3333333333333333}} - \sin^{-1} \left(1 - x\right) \]
10. Step-by-step derivation

    1. unpow1/3 9.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\pi}^{3} \cdot 0.125}} - \sin^{-1} \left(1 - x\right) \]

11. Simplified 9.7%

    \[\leadsto \color{blue}{\sqrt[3]{{\pi}^{3} \cdot 0.125}} - \sin^{-1} \left(1 - x\right) \]
12. Add Preprocessing

Alternative 6: 9.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 5.5e-17) (acos x) (acos (- 1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $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. Add Preprocessing

   3. Taylor expanded in x around inf 6.7%

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

      1. neg-mul-1 6.7%

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

   5. Simplified 6.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.7%

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

      3. sqr-neg 6.7%

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

      4. sqrt-unprod 6.7%

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

      5. add-sqr-sqrt 6.7%

         \[\leadsto \cos^{-1} \color{blue}{x} \]

      6. *-un-lft-identity 6.7%

         \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

   7. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.7%

         \[\leadsto \color{blue}{\cos^{-1} x} \]

   9. Simplified 6.7%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 74.9%

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

Alternative 7: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[x], $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 6.9%

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

   1. neg-mul-1 6.9%

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

5. Simplified 6.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 6.9%

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

   3. sqr-neg 6.9%

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

   4. sqrt-unprod 6.9%

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

   5. add-sqr-sqrt 6.9%

      \[\leadsto \cos^{-1} \color{blue}{x} \]

   6. *-un-lft-identity 6.9%

      \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]

7. Applied egg-rr 6.9%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} x} \]
8. Step-by-step derivation

   1. *-lft-identity 6.9%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

9. Simplified 6.9%

   \[\leadsto \color{blue}{\cos^{-1} x} \]
10. Add Preprocessing

Alternative 8: 3.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Add Preprocessing

Developer Target 1: 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 2024139 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))