Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.3% → 99.3%

Time: 20.3s

Alternatives: 7

Speedup: 1.0×

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: 99.3% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{-1}{{t\_0}^{-2} \cdot \left(-t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (acos (/ (fma -5.0 (* v v) 1.0) (fma v v -1.0)))))
   (/ -1.0 (* (pow t_0 -2.0) (- t_0)))))

C

double code(double v) {
	double t_0 = acos((fma(-5.0, (v * v), 1.0) / fma(v, v, -1.0)));
	return -1.0 / (pow(t_0, -2.0) * -t_0);
}

Julia

function code(v)
	t_0 = acos(Float64(fma(-5.0, Float64(v * v), 1.0) / fma(v, v, -1.0)))
	return Float64(-1.0 / Float64((t_0 ^ -2.0) * Float64(-t_0)))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcCos[N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(-1.0 / N[(N[Power[t$95$0, -2.0], $MachinePrecision] * (-t$95$0)), $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

4. Applied rewrites 99.5%

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

   1. lift-PI.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. asin-acos N/A

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

   10. lift-acos.f64 N/A

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

   11. associate-+r- N/A

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

   12. lower--.f64 N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}{{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{-2}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{\cos^{-1} \left(\frac{v \cdot \color{blue}{\left(v \cdot -5\right)} + 1}{v \cdot v + -1}\right)}}{{\cos^{-1} \left(\frac{v \cdot \left(v \cdot -5\right) + 1}{v \cdot v + -1}\right)}^{-2}} \]

   2. lift-fma.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-acos.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-acos.f64 N/A

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

10. Applied rewrites 99.5%

    \[\leadsto \color{blue}{\frac{-1}{{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{-2} \cdot \left(-\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
11. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(5, v \cdot v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (fma PI 0.5 (asin (/ (fma 5.0 (* v v) -1.0) (fma v v -1.0)))))

C

double code(double v) {
	return fma(((double) M_PI), 0.5, asin((fma(5.0, (v * v), -1.0) / fma(v, v, -1.0))));
}

Julia

function code(v)
	return fma(pi, 0.5, asin(Float64(fma(5.0, Float64(v * v), -1.0) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(Pi * 0.5 + N[ArcSin[N[(N[(5.0 * N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision] / N[(v * v + -1.0), $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. 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. acos-asin N/A

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

   8. sub-neg N/A

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

   9. div-inv N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. metadata-eval N/A

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

   13. asin-neg N/A

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

   14. lift-/.f64 N/A

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

   15. distribute-frac-neg2 N/A

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[ArcCos[N[(N[(v * N[(-5.0 * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $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. 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 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 4: 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 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

   \[\leadsto \cos^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) - 1\right)} \]
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 98.7

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

5. Applied rewrites 98.7%

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

Alternative 5: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (fma PI 0.5 (- (asin (fma v (* v 4.0) -1.0)))))

C

double code(double v) {
	return fma(((double) M_PI), 0.5, -asin(fma(v, (v * 4.0), -1.0)));
}

Julia

function code(v)
	return fma(pi, 0.5, Float64(-asin(fma(v, Float64(v * 4.0), -1.0))))
end

Wolfram

code[v_] := N[(Pi * 0.5 + (-N[ArcSin[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $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 98.4

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

5. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. acos-asin N/A

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

   4. sub-neg N/A

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

   5. lift-PI.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-asin.f64 98.4

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

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)} \]
8. Add Preprocessing

Alternative 6: 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 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 98.4

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

5. Applied rewrites 98.4%

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

Alternative 7: 98.0% 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 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

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

5. Applied rewrites 97.8%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))