bug323 (missed optimization)

Percentage Accurate: 77.0% → 78.9%

Time: 31.5s

Alternatives: 7

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: 77.0% 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: 78.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\pi} \cdot \sqrt{0.5}, \sqrt{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.1e-84)
   (acos (- 1.0 x))
   (fma (* (sqrt PI) (sqrt 0.5)) (sqrt (* PI 0.5)) (- (asin (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.1e-84) {
		tmp = acos((1.0 - x));
	} else {
		tmp = fma((sqrt(((double) M_PI)) * sqrt(0.5)), sqrt((((double) M_PI) * 0.5)), -asin((1.0 - x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.1e-84)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = fma(Float64(sqrt(pi) * sqrt(0.5)), sqrt(Float64(pi * 0.5)), Float64(-asin(Float64(1.0 - x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.1e-84], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] + (-N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.10000000000000005e-84

   1. Initial program 100.0%

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

   if 4.10000000000000005e-84 < x

   1. Initial program 15.7%

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

      1. acos-asin 15.8%

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

      2. add-sqr-sqrt 14.4%

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

      3. fma-neg 14.4%

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

      4. div-inv 14.4%

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

      5. metadata-eval 14.4%

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

      6. div-inv 14.4%

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

      7. metadata-eval 14.4%

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

   4. Applied egg-rr 14.4%

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

      1. sqrt-prod 22.8%

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

   6. Applied egg-rr 22.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\pi} \cdot \sqrt{0.5}, \sqrt{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot {\left(\sqrt{0.5}\right)}^{2} - \sin^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.1e-84)
   (acos (- 1.0 x))
   (- (* PI (pow (sqrt 0.5) 2.0)) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.1e-84) {
		tmp = acos((1.0 - x));
	} else {
		tmp = (((double) M_PI) * pow(sqrt(0.5), 2.0)) - asin((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.1e-84) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = (Math.PI * Math.pow(Math.sqrt(0.5), 2.0)) - Math.asin((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.1e-84:
		tmp = math.acos((1.0 - x))
	else:
		tmp = (math.pi * math.pow(math.sqrt(0.5), 2.0)) - math.asin((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.1e-84)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = Float64(Float64(pi * (sqrt(0.5) ^ 2.0)) - asin(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.1e-84)
		tmp = acos((1.0 - x));
	else
		tmp = (pi * (sqrt(0.5) ^ 2.0)) - asin((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.1e-84], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(Pi * N[Power[N[Sqrt[0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.10000000000000005e-84

   1. Initial program 100.0%

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

   if 4.10000000000000005e-84 < x

   1. Initial program 15.7%

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

      1. acos-asin 15.8%

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

      2. add-sqr-sqrt 14.4%

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

      3. fma-neg 14.4%

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

      4. div-inv 14.4%

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

      5. metadata-eval 14.4%

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

      6. div-inv 14.4%

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

      7. metadata-eval 14.4%

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

   4. Applied egg-rr 14.4%

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

   5. Taylor expanded in x around 0 22.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot {\left(\sqrt{0.5}\right)}^{2} - \sin^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{{\pi}^{3}} - \sin^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.1e-84)
   (acos (- 1.0 x))
   (- (* 0.5 (cbrt (pow PI 3.0))) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.1e-84) {
		tmp = acos((1.0 - x));
	} else {
		tmp = (0.5 * cbrt(pow(((double) M_PI), 3.0))) - asin((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.1e-84) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = (0.5 * Math.cbrt(Math.pow(Math.PI, 3.0))) - Math.asin((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.1e-84)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = Float64(Float64(0.5 * cbrt((pi ^ 3.0))) - asin(Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.1e-84], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.10000000000000005e-84

   1. Initial program 100.0%

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

   if 4.10000000000000005e-84 < x

   1. Initial program 15.7%

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

      1. acos-asin 15.8%

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

      2. sub-neg 15.8%

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

      3. div-inv 15.8%

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

      4. metadata-eval 15.8%

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

   4. Applied egg-rr 15.8%

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

      1. sub-neg 15.8%

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

   6. Simplified 15.8%

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

      1. add-cbrt-cube 22.4%

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

      2. pow3 22.4%

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

   8. Applied egg-rr 22.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{{\pi}^{3}} - \sin^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ 0.3333333333333333 \cdot \left(t_0 \cdot 2\right) + \left(\left(1 + t_0 \cdot 0.3333333333333333\right) + -1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (+
    (* 0.3333333333333333 (* t_0 2.0))
    (+ (+ 1.0 (* t_0 0.3333333333333333)) -1.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	return (0.3333333333333333 * (t_0 * 2.0)) + ((1.0 + (t_0 * 0.3333333333333333)) + -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = acos((1.0d0 - x))
    code = (0.3333333333333333d0 * (t_0 * 2.0d0)) + ((1.0d0 + (t_0 * 0.3333333333333333d0)) + (-1.0d0))
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	return (0.3333333333333333 * (t_0 * 2.0)) + ((1.0 + (t_0 * 0.3333333333333333)) + -1.0);
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	return (0.3333333333333333 * (t_0 * 2.0)) + ((1.0 + (t_0 * 0.3333333333333333)) + -1.0)

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	return Float64(Float64(0.3333333333333333 * Float64(t_0 * 2.0)) + Float64(Float64(1.0 + Float64(t_0 * 0.3333333333333333)) + -1.0))
end

MATLAB

function tmp = code(x)
	t_0 = acos((1.0 - x));
	tmp = (0.3333333333333333 * (t_0 * 2.0)) + ((1.0 + (t_0 * 0.3333333333333333)) + -1.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.3333333333333333 * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.3%

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

   1. add-log-exp 72.3%

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

   2. add-cube-cbrt 72.3%

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

   3. log-prod 72.3%

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

   4. cbrt-unprod 72.3%

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

   5. pow2 72.3%

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

4. Applied egg-rr 72.3%

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

   1. expm1-log1p-u 72.3%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)\right)} \]

   2. expm1-udef 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\sqrt[3]{e^{\cos^{-1} \left(1 - x\right)}}\right)\right)} - 1\right)} \]

   3. pow1/3 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}\right)} - 1\right) \]

   4. log-pow 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(1 - x\right)}\right)}\right)} - 1\right) \]

   5. add-log-exp 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(1 - x\right)}\right)} - 1\right) \]

6. Applied egg-rr 72.4%

   \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right)} - 1\right)} \]
7. Step-by-step derivation

   1. sub-neg 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right)} + \left(-1\right)\right)} \]

   2. log1p-udef 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \left(e^{\color{blue}{\log \left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right)}} + \left(-1\right)\right) \]

   3. rem-exp-log 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \left(\color{blue}{\left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right)} + \left(-1\right)\right) \]

   4. metadata-eval 72.4%

      \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \left(\left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) + \color{blue}{-1}\right) \]

8. Applied egg-rr 72.4%

   \[\leadsto \log \left(\sqrt[3]{{\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}}\right) + \color{blue}{\left(\left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) + -1\right)} \]
9. Step-by-step derivation

   1. pow1/3 72.4%

      \[\leadsto \log \color{blue}{\left({\left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}\right)}^{0.3333333333333333}\right)} + \left(\left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) + -1\right) \]

   2. log-pow 72.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left({\left(e^{\cos^{-1} \left(1 - x\right)}\right)}^{2}\right)} + \left(\left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) + -1\right) \]

   3. pow-exp 72.4%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left(e^{\cos^{-1} \left(1 - x\right) \cdot 2}\right)} + \left(\left(1 + 0.3333333333333333 \cdot \cos^{-1} \left(1 - x\right)\right) + -1\right) \]

   4. rem-log-exp 72.4%

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

10. Applied egg-rr 72.4%

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

11. Final simplification 72.4%

    \[\leadsto 0.3333333333333333 \cdot \left(\cos^{-1} \left(1 - x\right) \cdot 2\right) + \left(\left(1 + \cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right) + -1\right) \]
12. Add Preprocessing

Alternative 5: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. acos-asin 72.4%

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

   2. sub-neg 72.4%

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

   3. div-inv 72.4%

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

   4. metadata-eval 72.4%

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

4. Applied egg-rr 72.4%

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

   1. sub-neg 72.4%

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

6. Simplified 72.4%

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

   1. metadata-eval 72.4%

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

   2. div-inv 72.4%

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

   3. expm1-log1p-u 72.4%

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

   4. acos-asin 72.4%

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

   5. expm1-udef 72.3%

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

   6. log1p-udef 72.3%

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

   7. add-exp-log 72.3%

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

   8. associate--l+ 72.4%

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

   9. +-commutative 72.4%

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

   10. sub-neg 72.4%

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

   11. metadata-eval 72.4%

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

8. Applied egg-rr 72.4%

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

9. Final simplification 72.4%

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

Alternative 6: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (((double) M_PI) * 0.5) - asin((1.0 - x));
}

Java

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

Python

def code(x):
	return (math.pi * 0.5) - math.asin((1.0 - x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi * 0.5) - asin((1.0 - x));
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.3%

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

   1. acos-asin 72.4%

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

   2. sub-neg 72.4%

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

   3. div-inv 72.4%

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

   4. metadata-eval 72.4%

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

4. Applied egg-rr 72.4%

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

   1. sub-neg 72.4%

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

6. Simplified 72.4%

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

7. Final simplification 72.4%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(1 - x\right) \]
8. Add Preprocessing

Alternative 7: 77.0% 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 72.3%

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

3. Final simplification 72.3%

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

Developer target: 28.2% 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 2024018 
(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)))