Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 17.7s

Alternatives: 5

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: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\\ t_1 := \sin^{-1} t\_0\\ \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\left(\mathsf{fma}\left(\pi, 0.5, -\cos^{-1} t\_0\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\pi, 0.5, t\_1\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))) (t_1 (asin t_0)))
   (*
    (- (* (* PI (* PI PI)) 0.125) (pow (fma PI 0.5 (- (acos t_0))) 3.0))
    (/ 1.0 (fma t_1 (fma PI 0.5 t_1) (* (* PI PI) 0.25))))))

C

double code(double v) {
	double t_0 = fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0);
	double t_1 = asin(t_0);
	return (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.125) - pow(fma(((double) M_PI), 0.5, -acos(t_0)), 3.0)) * (1.0 / fma(t_1, fma(((double) M_PI), 0.5, t_1), ((((double) M_PI) * ((double) M_PI)) * 0.25)));
}

Julia

function code(v)
	t_0 = Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))
	t_1 = asin(t_0)
	return Float64(Float64(Float64(Float64(pi * Float64(pi * pi)) * 0.125) - (fma(pi, 0.5, Float64(-acos(t_0))) ^ 3.0)) * Float64(1.0 / fma(t_1, fma(pi, 0.5, t_1), Float64(Float64(pi * pi) * 0.25))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[t$95$0], $MachinePrecision]}, N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[N[(Pi * 0.5 + (-N[ArcCos[t$95$0], $MachinePrecision])), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$1 * N[(Pi * 0.5 + t$95$1), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.8%

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{v \cdot \color{blue}{\left(v \cdot -5\right)} + 1}{v \cdot v + -1}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}{v \cdot v + -1}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   5. asin-acos N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   6. lift-acos.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   7. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   9. div-inv N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   11. lift-acos.f64 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   12. acos-asin N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   13. lift-PI.f64 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2} - \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   14. div-inv N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

   16. lift-asin.f64 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{8} - {\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} - \color{blue}{\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)\right)\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)} \]

6. Applied egg-rr 98.9%

   \[\leadsto \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, -\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \]
7. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   5. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   7. lift-acos.f64 98.8

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 98.1

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

5. Simplified 98.1%

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

Alternative 4: 98.7% 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 98.8%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 97.9

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

5. Simplified 97.9%

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

Alternative 5: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

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

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in v around 0

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

5. Simplified 97.1%

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

Reproduce

herbie shell --seed 2024210 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))