bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.5%

Time: 8.0s

Alternatives: 9

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: 7.0% 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: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-t\_1, t\_1, t\_0\right)\\ t\_2 + \left(\cos^{-1} \left(1 - x\right) + t\_2\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)) (t_2 (fma (- t_1) t_1 t_0)))
   (+ t_2 (+ (acos (- 1.0 x)) t_2))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	double t_2 = fma(-t_1, t_1, t_0);
	return t_2 + (acos((1.0 - x)) + t_2);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-t_1), t_1, t_0)
	return Float64(t_2 + Float64(acos(Float64(1.0 - x)) + t_2))
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]}, Block[{t$95$2 = N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]}, N[(t$95$2 + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 7.4%

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

   1. acos-asin 7.4%

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

   2. *-un-lft-identity 7.4%

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

   3. add-sqr-sqrt 10.6%

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

   4. prod-diff 10.6%

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

   5. add-sqr-sqrt 10.6%

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

   6. fma-neg 10.6%

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

   7. *-un-lft-identity 10.6%

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

   8. acos-asin 10.6%

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

   9. add-sqr-sqrt 10.6%

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

4. Applied egg-rr 10.6%

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

   1. acos-asin 7.4%

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

   2. *-un-lft-identity 7.4%

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

   3. add-sqr-sqrt 10.6%

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

   4. prod-diff 10.6%

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

   5. add-sqr-sqrt 10.6%

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

   6. fma-neg 10.6%

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

   7. *-un-lft-identity 10.6%

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

   8. acos-asin 10.6%

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

   9. add-sqr-sqrt 10.6%

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

6. Applied egg-rr 10.6%

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

7. Final simplification 10.6%

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

Alternative 2: 10.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := 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. Initial program 7.4%

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

   1. acos-asin 7.4%

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

   2. add-sqr-sqrt 5.6%

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

   3. fma-neg 5.6%

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

   4. div-inv 5.6%

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

   5. metadata-eval 5.6%

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

   6. div-inv 5.6%

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

   7. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

   1. sqrt-prod 10.6%

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

6. Applied egg-rr 10.6%

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

Alternative 3: 10.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot {\left(\sqrt{0.5}\right)}^{2} + \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* PI (pow (sqrt 0.5) 2.0)) (- (acos (- 1.0 x)) (* PI 0.5))))

C

double code(double x) {
	return (((double) M_PI) * pow(sqrt(0.5), 2.0)) + (acos((1.0 - x)) - (((double) M_PI) * 0.5));
}

Java

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

Python

def code(x):
	return (math.pi * math.pow(math.sqrt(0.5), 2.0)) + (math.acos((1.0 - x)) - (math.pi * 0.5))

Julia

function code(x)
	return Float64(Float64(pi * (sqrt(0.5) ^ 2.0)) + Float64(acos(Float64(1.0 - x)) - Float64(pi * 0.5)))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(Pi * N[Power[N[Sqrt[0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

   1. acos-asin 7.4%

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

   2. add-sqr-sqrt 5.6%

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

   3. fma-neg 5.6%

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

   4. div-inv 5.6%

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

   5. metadata-eval 5.6%

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

   6. div-inv 5.6%

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

   7. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

5. Taylor expanded in x around 0 10.6%

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

   1. asin-acos 10.6%

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

   2. div-inv 10.6%

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

   3. metadata-eval 10.6%

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

7. Applied egg-rr 10.6%

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

8. Final simplification 10.6%

   \[\leadsto \pi \cdot {\left(\sqrt{0.5}\right)}^{2} + \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right) \]
9. Add Preprocessing

Alternative 4: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos x) (cbrt (pow (acos (- 1.0 x)) 3.0))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(x);
	} else {
		tmp = cbrt(pow(acos((1.0 - x)), 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(x);
	} else {
		tmp = Math.cbrt(Math.pow(Math.acos((1.0 - x)), 3.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(x);
	else
		tmp = cbrt((acos(Float64(1.0 - x)) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[x], $MachinePrecision], N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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. Add Preprocessing

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. *-un-lft-identity 6.5%

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

      2. *-commutative 6.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.5%

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

      5. sqr-neg 6.5%

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

      6. sqrt-unprod 6.5%

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

      7. add-sqr-sqrt 6.5%

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

   7. Applied egg-rr 6.5%

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

      1. *-rgt-identity 6.5%

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

   9. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.1%

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

      1. add-cbrt-cube 60.1%

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

      2. pow3 60.2%

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

   4. Applied egg-rr 60.2%

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

Alternative 5: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := 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. Initial program 7.4%

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

   1. acos-asin 7.4%

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

   2. add-sqr-sqrt 5.6%

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

   3. fma-neg 5.6%

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

   4. div-inv 5.6%

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

   5. metadata-eval 5.6%

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

   6. div-inv 5.6%

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

   7. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

5. Taylor expanded in x around 0 10.6%

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

Alternative 6: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. acos-asin 7.4%

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

   2. add-sqr-sqrt 5.6%

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

   3. fma-neg 5.6%

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

   4. div-inv 5.6%

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

   5. metadata-eval 5.6%

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

   6. div-inv 5.6%

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

   7. metadata-eval 5.6%

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

4. Applied egg-rr 5.6%

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

5. Taylor expanded in x around 0 10.6%

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

   1. add-cbrt-cube 10.5%

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

   2. pow3 10.5%

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

   3. sqrt-pow2 10.6%

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

   4. metadata-eval 10.6%

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

   5. metadata-eval 10.6%

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

7. Applied egg-rr 10.6%

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

Alternative 7: 9.6% 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 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

      1. *-un-lft-identity 6.5%

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

      2. *-commutative 6.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 6.5%

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

      5. sqr-neg 6.5%

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

      6. sqrt-unprod 6.5%

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

      7. add-sqr-sqrt 6.5%

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

   7. Applied egg-rr 6.5%

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

      1. *-rgt-identity 6.5%

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

   9. Simplified 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 60.1%

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

Alternative 8: 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 7.4%

   \[\cos^{-1} \left(1 - x\right) \]
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. *-un-lft-identity 6.9%

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

   2. *-commutative 6.9%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 6.9%

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

   5. sqr-neg 6.9%

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

   6. sqrt-unprod 6.9%

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

   7. add-sqr-sqrt 6.9%

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

7. Applied egg-rr 6.9%

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

   1. *-rgt-identity 6.9%

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

9. Simplified 6.9%

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

Alternative 9: 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 7.4%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024116 
(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)))