bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.5%

Time: 14.1s

Alternatives: 7

Speedup: 0.9×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\\ t_1 := -\frac{{\sin^{-1} 1}^{2}}{t\_0}\\ t_2 := \frac{0.25 \cdot \left(\pi \cdot \pi\right)}{t\_0}\\ t_3 := \mathsf{fma}\left(t\_1, t\_1 - t\_2, 0.0625 \cdot {\left(\frac{t\_0}{\pi \cdot \pi}\right)}^{-2}\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{{t\_2}^{3} \cdot t\_3 - t\_3 \cdot {\left(t\_0 \cdot {\sin^{-1} 1}^{-2}\right)}^{-3}}{{t\_3}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI 0.5 (asin 1.0)))
        (t_1 (- (/ (pow (asin 1.0) 2.0) t_0)))
        (t_2 (/ (* 0.25 (* PI PI)) t_0))
        (t_3 (fma t_1 (- t_1 t_2) (* 0.0625 (pow (/ t_0 (* PI PI)) -2.0)))))
   (if (<= x 5.5e-17)
     (/
      (-
       (* (pow t_2 3.0) t_3)
       (* t_3 (pow (* t_0 (pow (asin 1.0) -2.0)) -3.0)))
      (pow t_3 2.0))
     (acos (- 1.0 x)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), 0.5, asin(1.0));
	double t_1 = -(pow(asin(1.0), 2.0) / t_0);
	double t_2 = (0.25 * (((double) M_PI) * ((double) M_PI))) / t_0;
	double t_3 = fma(t_1, (t_1 - t_2), (0.0625 * pow((t_0 / (((double) M_PI) * ((double) M_PI))), -2.0)));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((pow(t_2, 3.0) * t_3) - (t_3 * pow((t_0 * pow(asin(1.0), -2.0)), -3.0))) / pow(t_3, 2.0);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, 0.5, asin(1.0))
	t_1 = Float64(-Float64((asin(1.0) ^ 2.0) / t_0))
	t_2 = Float64(Float64(0.25 * Float64(pi * pi)) / t_0)
	t_3 = fma(t_1, Float64(t_1 - t_2), Float64(0.0625 * (Float64(t_0 / Float64(pi * pi)) ^ -2.0)))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(Float64((t_2 ^ 3.0) * t_3) - Float64(t_3 * (Float64(t_0 * (asin(1.0) ^ -2.0)) ^ -3.0))) / (t_3 ^ 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]}, Block[{t$95$1 = (-N[(N[Power[N[ArcSin[1.0], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision])}, Block[{t$95$2 = N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$1 - t$95$2), $MachinePrecision] + N[(0.0625 * N[Power[N[(t$95$0 / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(t$95$3 * N[Power[N[(t$95$0 * N[Power[N[ArcSin[1.0], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. acos-asin N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied egg-rr 7.8%

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

   9. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\pi \cdot \left(\pi \cdot 0.25\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right)}^{3} - {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right) \cdot {\sin^{-1} 1}^{-2}\right)}^{-3}}{\mathsf{fma}\left(0.0625, {\left(\frac{\pi \cdot \pi}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right)}^{2}, \mathsf{fma}\left({\sin^{-1} 1}^{4}, {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}^{-2}, \frac{\pi \cdot \left(\pi \cdot 0.25\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)} \cdot \frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right)\right)}} \]

   11. Applied egg-rr 7.8%

       \[\leadsto \color{blue}{\frac{{\left(\frac{0.25 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right)}^{3} \cdot \mathsf{fma}\left(-\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, \left(-\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right) - \frac{0.25 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, 0.0625 \cdot {\left(\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}{\pi \cdot \pi}\right)}^{-2}\right) - \mathsf{fma}\left(-\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, \left(-\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right) - \frac{0.25 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, 0.0625 \cdot {\left(\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}{\pi \cdot \pi}\right)}^{-2}\right) \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right) \cdot {\sin^{-1} 1}^{-2}\right)}^{-3}}{{\left(\mathsf{fma}\left(-\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, \left(-\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right) - \frac{0.25 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, 0.0625 \cdot {\left(\frac{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}{\pi \cdot \pi}\right)}^{-2}\right)\right)}^{2}}} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 68.5%

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

Alternative 2: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, t\_1, t\_1 \cdot \left(-{\sin^{-1} 1}^{3}\right)\right)}{{t\_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))
        (t_1 (fma PI (* 0.25 PI) (* (asin 1.0) (fma PI 0.5 (asin 1.0))))))
   (if (<= t_0 0.0)
     (/
      (fma (* (* PI (* PI PI)) 0.125) t_1 (* t_1 (- (pow (asin 1.0) 3.0))))
      (pow t_1 2.0))
     t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = fma(((double) M_PI), (0.25 * ((double) M_PI)), (asin(1.0) * fma(((double) M_PI), 0.5, asin(1.0))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.125), t_1, (t_1 * -pow(asin(1.0), 3.0))) / pow(t_1, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = fma(pi, Float64(0.25 * pi), Float64(asin(1.0) * fma(pi, 0.5, asin(1.0))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(pi * Float64(pi * pi)) * 0.125), t_1, Float64(t_1 * Float64(-(asin(1.0) ^ 3.0)))) / (t_1 ^ 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.25 * Pi), $MachinePrecision] + N[(N[ArcSin[1.0], $MachinePrecision] * N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$1 + N[(t$95$1 * (-N[Power[N[ArcSin[1.0], $MachinePrecision], 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $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 3.8%

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

      1. acos-asin N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied egg-rr 7.8%

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

   9. Applied egg-rr 7.8%

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

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

   1. Initial program 68.5%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125, \mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right), \mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right) \cdot \left(-{\sin^{-1} 1}^{3}\right)\right)}{{\left(\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} 1 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 10.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\pi \cdot \left(0.25 \cdot \pi\right), t\_0, -t\_0 \cdot {\sin^{-1} 1}^{2}\right)}{{t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI 0.5 (asin 1.0))))
   (if (<= x 5.5e-17)
     (/
      (fma (* PI (* 0.25 PI)) 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 <= 5.5e-17) {
		tmp = fma((((double) M_PI) * (0.25 * ((double) M_PI))), 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 <= 5.5e-17)
		tmp = Float64(fma(Float64(pi * Float64(0.25 * pi)), t_0, Float64(-Float64(t_0 * (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, 5.5e-17], N[(N[(N[(Pi * N[(0.25 * Pi), $MachinePrecision]), $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 < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. acos-asin N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied egg-rr 7.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-asin.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-asin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-asin.f64 N/A

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

      13. lift-fma.f64 N/A

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

   9. Applied egg-rr 7.8%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 68.5%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\pi \cdot \left(0.25 \cdot \pi\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), -\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right) \cdot {\sin^{-1} 1}^{2}\right)}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 10.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{t\_1}, \pi \cdot \pi, -\frac{{\sin^{-1} 1}^{2}}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (fma PI 0.5 (asin 1.0))))
   (if (<= t_0 0.0)
     (fma (/ 0.25 t_1) (* PI PI) (- (/ (pow (asin 1.0) 2.0) t_1)))
     t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = fma(((double) M_PI), 0.5, asin(1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma((0.25 / t_1), (((double) M_PI) * ((double) M_PI)), -(pow(asin(1.0), 2.0) / t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = fma(pi, 0.5, asin(1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(Float64(0.25 / t_1), Float64(pi * pi), Float64(-Float64((asin(1.0) ^ 2.0) / t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(0.25 / t$95$1), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + (-N[(N[Power[N[ArcSin[1.0], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision])), $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 3.8%

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

      1. acos-asin N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied egg-rr 7.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-asin.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. /-rgt-identity N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lift-asin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-asin.f64 N/A

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

   9. Applied egg-rr 7.8%

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

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

   1. Initial program 68.5%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}, \pi \cdot \pi, -\frac{{\sin^{-1} 1}^{2}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 9.5% accurate, 0.9× speedup

Math

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

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(Float64(-x));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.7

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

   5. Simplified 6.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 68.5%

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

Alternative 6: 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 7.9%

   \[\cos^{-1} \left(1 - x\right) \]
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. lower-neg.f64 7.1

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

5. Simplified 7.1%

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

Alternative 7: 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 7.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
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 2024207 
(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)))