bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.3%

Time: 9.7s

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: 6.7% 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: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, t\_0, t\_0 \cdot \left(-{\sin^{-1} 1}^{3}\right)\right)}{{t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI (* PI 0.25) (* (asin 1.0) (fma PI 0.5 (asin 1.0))))))
   (if (<= x 5.6e-17)
     (/
      (fma (* (* PI (* PI PI)) 0.125) t_0 (* t_0 (- (pow (asin 1.0) 3.0))))
      (pow t_0 2.0))
     (acos (- 1.0 x)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), (((double) M_PI) * 0.25), (asin(1.0) * fma(((double) M_PI), 0.5, asin(1.0))));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = fma(((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.125), t_0, (t_0 * -pow(asin(1.0), 3.0))) / pow(t_0, 2.0);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, Float64(pi * 0.25), Float64(asin(1.0) * fma(pi, 0.5, asin(1.0))))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(fma(Float64(Float64(pi * Float64(pi * pi)) * 0.125), t_0, Float64(t_0 * Float64(-(asin(1.0) ^ 3.0)))) / (t_0 ^ 2.0));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(N[ArcSin[1.0], $MachinePrecision] * N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$0 + N[(t$95$0 * (-N[Power[N[ArcSin[1.0], $MachinePrecision], 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.9%

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

      1. acos-asin N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} - \frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \frac{1}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)} \]
   8. Step-by-step derivation

      1. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}} + \color{blue}{\frac{\mathsf{neg}\left({\sin^{-1} 1}^{3}\right)}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} \]

      3. frac-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}\right) \cdot \left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) + \left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left({\sin^{-1} 1}^{3}\right)\right)}{\left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot \left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}} \]

   9. Applied egg-rr 7.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right), \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right) \cdot \left(-{\sin^{-1} 1}^{3}\right)\right)}{{\left(\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\right)}^{2}}} \]

   if 5.5999999999999998e-17 < x

   1. Initial program 68.8%

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

Alternative 2: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{t\_0}, \pi \cdot \left(\pi \cdot \pi\right), \frac{-{\sin^{-1} 1}^{3}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI (* PI 0.25) (* (asin 1.0) (fma PI 0.5 (asin 1.0))))))
   (if (<= x 5.6e-17)
     (fma (/ 0.125 t_0) (* PI (* PI PI)) (/ (- (pow (asin 1.0) 3.0)) t_0))
     (acos (- 1.0 x)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), (((double) M_PI) * 0.25), (asin(1.0) * fma(((double) M_PI), 0.5, asin(1.0))));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = fma((0.125 / t_0), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (-pow(asin(1.0), 3.0) / t_0));
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, Float64(pi * 0.25), Float64(asin(1.0) * fma(pi, 0.5, asin(1.0))))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = fma(Float64(0.125 / t_0), Float64(pi * Float64(pi * pi)), Float64(Float64(-(asin(1.0) ^ 3.0)) / t_0));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(N[ArcSin[1.0], $MachinePrecision] * N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(N[(0.125 / t$95$0), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[N[ArcSin[1.0], $MachinePrecision], 3.0], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.9%

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

      1. acos-asin N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} - \frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \frac{1}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{8} \cdot \frac{1}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \frac{1}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right)} \]

   9. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}, \pi \cdot \left(\pi \cdot \pi\right), -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}\right)} \]

   if 5.5999999999999998e-17 < x

   1. Initial program 68.8%

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

4. Final simplification 9.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.125}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}, \pi \cdot \left(\pi \cdot \pi\right), \frac{-{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{t\_0}, \frac{-{\sin^{-1} 1}^{3}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI (* PI 0.25) (* (asin 1.0) (fma PI 0.5 (asin 1.0))))))
   (if (<= x 5.6e-17)
     (fma PI (/ (* (* PI PI) 0.125) t_0) (/ (- (pow (asin 1.0) 3.0)) t_0))
     (acos (- 1.0 x)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), (((double) M_PI) * 0.25), (asin(1.0) * fma(((double) M_PI), 0.5, asin(1.0))));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = fma(((double) M_PI), (((((double) M_PI) * ((double) M_PI)) * 0.125) / t_0), (-pow(asin(1.0), 3.0) / t_0));
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, Float64(pi * 0.25), Float64(asin(1.0) * fma(pi, 0.5, asin(1.0))))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = fma(pi, Float64(Float64(Float64(pi * pi) * 0.125) / t_0), Float64(Float64(-(asin(1.0) ^ 3.0)) / t_0));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(N[ArcSin[1.0], $MachinePrecision] * N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(Pi * N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.125), $MachinePrecision] / t$95$0), $MachinePrecision] + N[((-N[Power[N[ArcSin[1.0], $MachinePrecision], 3.0], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.9%

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

      1. acos-asin N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} - \frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \frac{1}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)} \]
   8. Step-by-step derivation

      1. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8}\right)}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}, \mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right)} \]

   9. Applied egg-rr 7.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}\right)} \]

   if 5.5999999999999998e-17 < x

   1. Initial program 68.8%

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

4. Final simplification 9.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}, \frac{-{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 0.25, t\_0, t\_0 \cdot \left(-{\sin^{-1} 1}^{2}\right)\right)}{{t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI 0.5 (asin 1.0))))
   (if (<= x 5.6e-17)
     (/
      (fma (* (* PI PI) 0.25) t_0 (* t_0 (- (pow (asin 1.0) 2.0))))
      (pow t_0 2.0))
     (acos (- 1.0 x)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), 0.5, asin(1.0));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * 0.25), t_0, (t_0 * -pow(asin(1.0), 2.0))) / pow(t_0, 2.0);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, 0.5, asin(1.0))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = Float64(fma(Float64(Float64(pi * pi) * 0.25), t_0, Float64(t_0 * Float64(-(asin(1.0) ^ 2.0)))) / (t_0 ^ 2.0));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0 + N[(t$95$0 * (-N[Power[N[ArcSin[1.0], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.9%

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

      1. acos-asin N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} - \frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \frac{1}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)} \]

   9. Applied egg-rr 7.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 68.8%

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

Alternative 5: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\\ \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{t\_0}, \pi \cdot \pi, \frac{{\sin^{-1} 1}^{2}}{-t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI 0.5 (asin 1.0))))
   (if (<= x 5.6e-17)
     (fma (/ 0.25 t_0) (* PI PI) (/ (pow (asin 1.0) 2.0) (- t_0)))
     (acos (- 1.0 x)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), 0.5, asin(1.0));
	double tmp;
	if (x <= 5.6e-17) {
		tmp = fma((0.25 / t_0), (((double) M_PI) * ((double) M_PI)), (pow(asin(1.0), 2.0) / -t_0));
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, 0.5, asin(1.0))
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = fma(Float64(0.25 / t_0), Float64(pi * pi), Float64((asin(1.0) ^ 2.0) / Float64(-t_0)));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.6e-17], N[(N[(0.25 / t$95$0), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[Power[N[ArcSin[1.0], $MachinePrecision], 2.0], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.9%

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

      1. acos-asin N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} - \frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \frac{1}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)} \]
   8. Step-by-step derivation

      1. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}\right) \cdot \frac{1}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}} - \frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} - \frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}} \]

      3. sub-div N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} \]

   9. Applied egg-rr 7.5%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 68.8%

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

Alternative 6: 9.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= t_0 0.0) (acos (- x)) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (t_0 <= 0.0d0) then
        tmp = acos(-x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = acos(Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = acos(-x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[ArcCos[(-x)], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 6.6

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

   5. Simplified 6.6%

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

   if 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 68.8%

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

Alternative 7: 10.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\pi}}\\ \mathbf{if}\;1 - x \leq 0.9999999999999999:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot \sqrt{\pi}\right) \cdot t\_0, t\_0, -\sin^{-1} 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (if (<= (- 1.0 x) 0.9999999999999999)
     (acos (- 1.0 x))
     (fma (* (* 0.5 (sqrt PI)) t_0) t_0 (- (asin 1.0))))))

C

double code(double x) {
	double t_0 = sqrt(sqrt(((double) M_PI)));
	double tmp;
	if ((1.0 - x) <= 0.9999999999999999) {
		tmp = acos((1.0 - x));
	} else {
		tmp = fma(((0.5 * sqrt(((double) M_PI))) * t_0), t_0, -asin(1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(sqrt(pi))
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999999)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = fma(Float64(Float64(0.5 * sqrt(pi)) * t_0), t_0, Float64(-asin(1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999999], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(N[(0.5 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + (-N[ArcSin[1.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.999999999999999889

   1. Initial program 68.8%

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

   if 0.999999999999999889 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.9%

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

      1. acos-asin N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} - \frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} 1 \cdot \sin^{-1} 1 + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} 1\right)}\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \frac{1}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)} \]
   8. Step-by-step derivation

      1. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} + \left(\mathsf{neg}\left(\frac{{\sin^{-1} 1}^{3}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}\right)\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}} + \color{blue}{\frac{\mathsf{neg}\left({\sin^{-1} 1}^{3}\right)}{\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}} \]

      3. frac-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}\right) \cdot \left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) + \left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left({\sin^{-1} 1}^{3}\right)\right)}{\left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot \left(\sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right) + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}} \]

   9. Applied egg-rr 7.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right), \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right) \cdot \left(-{\sin^{-1} 1}^{3}\right)\right)}{{\left(\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) + \sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) + \sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right)\right) \cdot {\sin^{-1} 1}^{3}\right)\right)}}{{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) + \sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right)\right)}^{2}} \]

       2. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) + \sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right)\right) - \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) + \sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right)\right) \cdot {\sin^{-1} 1}^{3}}}{{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) + \sin^{-1} 1 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \sin^{-1} 1\right)\right)}^{2}} \]

   11. Applied egg-rr 7.5%

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

Alternative 8: 6.9% accurate, 1.0× speedup

Math

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

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(Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 6.8

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

5. Simplified 6.8%

   \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]
6. 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 6.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 3.8%

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

Developer Target 1: 100.0% accurate, 0.8× 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 2024199 
(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)))