bug323 (missed optimization)

Percentage Accurate: 7.2% → 10.7%

Time: 7.7s

Alternatives: 6

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: 7.2% 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: 10.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{t\_0}, -\frac{{\sin^{-1} 1}^{3}}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (asin 1.0) (fma PI 0.5 (asin 1.0)) (* (* PI PI) 0.25))))
   (if (<= x 5.5e-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(asin(1.0), fma(((double) M_PI), 0.5, asin(1.0)), ((((double) M_PI) * ((double) M_PI)) * 0.25));
	double tmp;
	if (x <= 5.5e-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(asin(1.0), fma(pi, 0.5, asin(1.0)), Float64(Float64(pi * pi) * 0.25))
	tmp = 0.0
	if (x <= 5.5e-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[(N[ArcSin[1.0], $MachinePrecision] * N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.5e-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 < 5.50000000000000001e-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. 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. accelerator-lowering-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 7.3%

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

   9. Applied egg-rr 7.3%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 55.6%

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

4. Final simplification 9.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}, -\frac{{\sin^{-1} 1}^{3}}{\mathsf{fma}\left(\sin^{-1} 1, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 10.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\\ \mathbf{if}\;1 - x \leq 0.9999999999999991:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 0.25, t\_0, -t\_0 \cdot {\sin^{-1} 1}^{2}\right)}{{t\_0}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI 0.5 (asin 1.0))))
   (if (<= (- 1.0 x) 0.9999999999999991)
     (acos (- 1.0 x))
     (/
      (fma (* (* PI PI) 0.25) t_0 (- (* t_0 (pow (asin 1.0) 2.0))))
      (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = fma(((double) M_PI), 0.5, asin(1.0));
	double tmp;
	if ((1.0 - x) <= 0.9999999999999991) {
		tmp = acos((1.0 - x));
	} else {
		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);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(pi, 0.5, asin(1.0))
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999991)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = Float64(fma(Float64(Float64(pi * pi) * 0.25), t_0, Float64(-Float64(t_0 * (asin(1.0) ^ 2.0)))) / (t_0 ^ 2.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi * 0.5 + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999991], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 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]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999911

   1. Initial program 55.6%

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

   if 0.99999999999999911 < (-.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. 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. accelerator-lowering-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 7.3%

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

      1. un-div-inv N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

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

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

   9. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right) \cdot \left(-{\sin^{-1} 1}^{2}\right)\right)}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999991:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 0.25, \mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right), -\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right) \cdot {\sin^{-1} 1}^{2}\right)}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} 1\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 10.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma PI 0.5 (asin 1.0))))
   (if (<= x 5.5e-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 <= 5.5e-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 <= 5.5e-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, 5.5e-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 < 5.50000000000000001e-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. 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. accelerator-lowering-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 7.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

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

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

      7. PI-lowering-PI.f64 N/A

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

      8. asin-lowering-asin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 7.3%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 55.6%

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

4. Final simplification 9.5%

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

Alternative 4: 9.8% 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 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.4

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

   5. Simplified 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 55.6%

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

Alternative 5: 7.0% 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 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.7

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

5. Simplified 6.7%

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

Alternative 6: 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 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
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. 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 2024204 
(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)))