bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.4%

Time: 7.6s

Alternatives: 10

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: 6.8% 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: 10.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (fma (* t_0 t_0) (* 0.5 (sqrt PI)) (- (fma PI 0.5 (- (acos (- 1.0 x))))))))

C

double code(double x) {
	double t_0 = sqrt(sqrt(((double) M_PI)));
	return fma((t_0 * t_0), (0.5 * sqrt(((double) M_PI))), -fma(((double) M_PI), 0.5, -acos((1.0 - x))));
}

Julia

function code(x)
	t_0 = sqrt(sqrt(pi))
	return fma(Float64(t_0 * t_0), Float64(0.5 * sqrt(pi)), Float64(-fma(pi, 0.5, Float64(-acos(Float64(1.0 - x))))))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.0

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

4. Applied rewrites 5.0%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. pow1/2 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. sqrt-pow2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 10.3

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

6. Applied rewrites 10.3%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 10.3

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

8. Applied rewrites 10.3%

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

9. Final simplification 10.3%

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

Alternative 2: 10.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (fma (* t_0 t_0) (* 0.5 (sqrt PI)) (- (asin (- 1.0 x))))))

C

double code(double x) {
	double t_0 = sqrt(sqrt(((double) M_PI)));
	return fma((t_0 * t_0), (0.5 * sqrt(((double) M_PI))), -asin((1.0 - x)));
}

Julia

function code(x)
	t_0 = sqrt(sqrt(pi))
	return fma(Float64(t_0 * t_0), Float64(0.5 * sqrt(pi)), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.0

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

4. Applied rewrites 5.0%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. pow1/2 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. sqrt-pow2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 10.3

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

6. Applied rewrites 10.3%

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

7. Final simplification 10.3%

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

Alternative 3: 10.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt PI))))
   (fma (sqrt PI) t_0 (fma t_0 (- (sqrt PI)) (acos (- 1.0 x))))))

C

double code(double x) {
	double t_0 = 0.5 * sqrt(((double) M_PI));
	return fma(sqrt(((double) M_PI)), t_0, fma(t_0, -sqrt(((double) M_PI)), acos((1.0 - x))));
}

Julia

function code(x)
	t_0 = Float64(0.5 * sqrt(pi))
	return fma(sqrt(pi), t_0, fma(t_0, Float64(-sqrt(pi)), acos(Float64(1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.0

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

4. Applied rewrites 5.0%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-neg.f64 10.2

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

6. Applied rewrites 10.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-neg-in N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. *-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. remove-double-neg N/A

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

8. Applied rewrites 10.2%

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

9. Final simplification 10.2%

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

Alternative 4: 10.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.0

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

4. Applied rewrites 5.0%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. pow1/2 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. sqrt-pow2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 10.3

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

6. Applied rewrites 10.3%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 10.3

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

8. Applied rewrites 10.3%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt 5.0

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. +-commutative N/A

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

   12. lift-neg.f64 N/A

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

   13. neg-mul-1 N/A

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

10. Applied rewrites 10.2%

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

11. Final simplification 10.2%

    \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt{\pi}, -\sqrt{\pi}, 0.5 \cdot \pi + \cos^{-1} \left(1 - x\right)\right) \]
12. Add Preprocessing

Alternative 5: 10.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999996:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(\pi, -0.5, \cos^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\pi}, 0.25 \cdot \left(\pi \cdot \pi\right), -\sin^{-1} 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 0.9999999999999996)
   (fma PI 0.5 (fma PI -0.5 (acos (- 1.0 x))))
   (fma (/ 2.0 PI) (* 0.25 (* PI PI)) (- (asin 1.0)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999996) {
		tmp = fma(((double) M_PI), 0.5, fma(((double) M_PI), -0.5, acos((1.0 - x))));
	} else {
		tmp = fma((2.0 / ((double) M_PI)), (0.25 * (((double) M_PI) * ((double) M_PI))), -asin(1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999996)
		tmp = fma(pi, 0.5, fma(pi, -0.5, acos(Float64(1.0 - x))));
	else
		tmp = fma(Float64(2.0 / pi), Float64(0.25 * Float64(pi * pi)), Float64(-asin(1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999996], N[(Pi * 0.5 + N[(Pi * -0.5 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / Pi), $MachinePrecision] * N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + (-N[ArcSin[1.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999956

   1. Initial program 61.5%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 61.4

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

   4. Applied rewrites 61.4%

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

      1. lift-asin.f64 N/A

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

      2. asin-acos N/A

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

      3. lift-acos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 61.5

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

   6. Applied rewrites 61.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 61.5

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

   8. Applied rewrites 61.5%

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

   if 0.99999999999999956 < (-.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. Applied rewrites 3.9%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lift-PI.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sub-neg N/A

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

   7. Applied rewrites 7.5%

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

4. Final simplification 10.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999996:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(\pi, -0.5, \cos^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\pi}, 0.25 \cdot \left(\pi \cdot \pi\right), -\sin^{-1} 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{2}{\pi}, 0.25 \cdot \left(\pi \cdot \pi\right), -\sin^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (/ 2.0 PI) (* 0.25 (* PI PI)) (- (asin (- 1.0 x)))))

C

double code(double x) {
	return fma((2.0 / ((double) M_PI)), (0.25 * (((double) M_PI) * ((double) M_PI))), -asin((1.0 - x)));
}

Julia

function code(x)
	return fma(Float64(2.0 / pi), Float64(0.25 * Float64(pi * pi)), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(N[(2.0 / Pi), $MachinePrecision] * N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + (-N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.0

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

4. Applied rewrites 5.0%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 10.2%

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

7. Final simplification 10.2%

   \[\leadsto \mathsf{fma}\left(\frac{2}{\pi}, 0.25 \cdot \left(\pi \cdot \pi\right), -\sin^{-1} \left(1 - x\right)\right) \]
8. Add Preprocessing

Alternative 7: 9.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos (- x)) (fma PI 0.5 (fma PI -0.5 (acos (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = fma(((double) M_PI), 0.5, fma(((double) M_PI), -0.5, acos((1.0 - x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = fma(pi, 0.5, fma(pi, -0.5, acos(Float64(1.0 - x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[(Pi * 0.5 + N[(Pi * -0.5 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $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. lower-neg.f64 6.5

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

   5. Applied rewrites 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 61.5%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 61.4

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

   4. Applied rewrites 61.4%

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

      1. lift-asin.f64 N/A

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

      2. asin-acos N/A

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

      3. lift-acos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 61.5

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

   6. Applied rewrites 61.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 61.5

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

   8. Applied rewrites 61.5%

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

Alternative 8: 9.4% 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. lower-neg.f64 6.5

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

   5. Applied rewrites 6.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 61.5%

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

Alternative 9: 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 6.8%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.8

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

5. Applied rewrites 6.8%

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

Alternative 10: 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 6.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 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 2024235 
(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)))