Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 98.6%

Time: 13.8s

Alternatives: 3

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\\ \frac{\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.125\right) - {t\_0}^{3}}{\mathsf{fma}\left(\pi \cdot t\_0, 0.5, \mathsf{fma}\left(\pi, \pi \cdot 0.25, {t\_0}^{2}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (fma v (* v 4.0) -1.0))))
   (/
    (- (* PI (* (* PI PI) 0.125)) (pow t_0 3.0))
    (fma (* PI t_0) 0.5 (fma PI (* PI 0.25) (pow t_0 2.0))))))

C

double code(double v) {
	double t_0 = asin(fma(v, (v * 4.0), -1.0));
	return ((((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * 0.125)) - pow(t_0, 3.0)) / fma((((double) M_PI) * t_0), 0.5, fma(((double) M_PI), (((double) M_PI) * 0.25), pow(t_0, 2.0)));
}

Julia

function code(v)
	t_0 = asin(fma(v, Float64(v * 4.0), -1.0))
	return Float64(Float64(Float64(pi * Float64(Float64(pi * pi) * 0.125)) - (t_0 ^ 3.0)) / fma(Float64(pi * t_0), 0.5, fma(pi, Float64(pi * 0.25), (t_0 ^ 2.0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * t$95$0), $MachinePrecision] * 0.5 + N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. unpow2 N/A

      \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(4 \cdot v\right) \cdot v} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{v \cdot \left(4 \cdot v\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(v \cdot \left(4 \cdot v\right) + \color{blue}{-1}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v, 4 \cdot v, -1\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, \color{blue}{v \cdot 4}, -1\right)\right) \]

   8. lower-*.f64 99.5

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, \color{blue}{v \cdot 4}, -1\right)\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)} \]
6. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)} \]

   2. acos-asin N/A

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

   3. flip3-- N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.125\right) - {\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}^{3}}{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.5, {\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}^{2} + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)}} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. swap-sqr N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

9. Applied rewrites 99.5%

   \[\leadsto \frac{\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.125\right) - {\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right) \cdot \pi, 0.5, \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}^{2}\right)\right)}} \]

10. Final simplification 99.5%

    \[\leadsto \frac{\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.125\right) - {\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}^{3}}{\mathsf{fma}\left(\pi \cdot \sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right), 0.5, \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}^{2}\right)\right)} \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (fma v (* v 4.0) -1.0)))

C

double code(double v) {
	return acos(fma(v, (v * 4.0), -1.0));
}

Julia

function code(v)
	return acos(fma(v, Float64(v * 4.0), -1.0))
end

Wolfram

code[v_] := N[ArcCos[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. unpow2 N/A

      \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(4 \cdot v\right) \cdot v} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{v \cdot \left(4 \cdot v\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(v \cdot \left(4 \cdot v\right) + \color{blue}{-1}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v, 4 \cdot v, -1\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, \color{blue}{v \cdot 4}, -1\right)\right) \]

   8. lower-*.f64 98.6

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, \color{blue}{v \cdot 4}, -1\right)\right) \]

5. Applied rewrites 98.6%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))