Falkner and Boettcher, Appendix B, 1

Average Accuracy: 99.1% → 99.1%

Time: 18.0s

Precision: binary64

Cost: 98496

Math

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

↓

\[\begin{array}{l} t_0 := \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ t_1 := \sqrt[3]{t_0 + 2}\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-1 + {\left(1 + t_0\right)}^{2}}{t_1}}{{t_1}^{2}}\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0 (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))
        (t_1 (cbrt (+ t_0 2.0))))
   (expm1 (log1p (/ (/ (+ -1.0 (pow (+ 1.0 t_0) 2.0)) t_1) (pow t_1 2.0))))))

C

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

↓

double code(double v) {
	double t_0 = acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)));
	double t_1 = cbrt((t_0 + 2.0));
	return expm1(log1p((((-1.0 + pow((1.0 + t_0), 2.0)) / t_1) / pow(t_1, 2.0))));
}

Julia

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

↓

function code(v)
	t_0 = acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))
	t_1 = cbrt(Float64(t_0 + 2.0))
	return expm1(log1p(Float64(Float64(Float64(-1.0 + (Float64(1.0 + t_0) ^ 2.0)) / t_1) / (t_1 ^ 2.0))))
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]

↓

code[v_] := Block[{t$95$0 = N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(t$95$0 + 2.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(Exp[N[Log[1 + N[(N[(N[(-1.0 + N[Power[N[(1.0 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Initial program 99.1%

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

2. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
   Proof

   [Start] 99.1	\[ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
   expm1-log1p-u [=>] 99.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)} \]
   sub-neg [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{v \cdot v - 1}\right)\right)\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) + 1}}{v \cdot v - 1}\right)\right)\right) \]
   *-commutative [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\left(-\color{blue}{\left(v \cdot v\right) \cdot 5}\right) + 1}{v \cdot v - 1}\right)\right)\right) \]
   distribute-rgt-neg-in [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{v \cdot v - 1}\right)\right)\right) \]
   fma-def [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{v \cdot v - 1}\right)\right)\right) \]
   metadata-eval [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{v \cdot v - 1}\right)\right)\right) \]
   fma-neg [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)\right) \]
   metadata-eval [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)\right)\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2} - 1}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + 1}}\right)\right) \]
   Proof

   [Start] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
   expm1-log1p-u [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}\right)\right) \]
   expm1-udef [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1}\right)\right) \]
   flip-- [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} + 1}}\right)\right) \]

4. Simplified 99.1%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)\right) \]
   Proof

   [Start] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2} - 1}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + 1}\right)\right) \]
   sub-neg [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2} + \left(-1\right)}}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + 1}\right)\right) \]
   metadata-eval [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2} + \color{blue}{-1}}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + 1}\right)\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + 1}\right)\right) \]
   +-commutative [<=] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\color{blue}{1 + \left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)\right) \]
   associate-+r+ [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\color{blue}{\left(1 + 1\right) + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)\right) \]
   metadata-eval [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\color{blue}{2} + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)}^{2}} \cdot \frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}}\right)\right) \]
   Proof

   [Start] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
   *-un-lft-identity [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}\right)}}{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
   add-cube-cbrt [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 \cdot \left(-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}\right)}{\color{blue}{\left(\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) \cdot \sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}\right)\right) \]
   times-frac [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot \frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}\right)\right) \]
   pow2 [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}}} \cdot \frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(\sqrt[3]{\color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}\right)}^{2}} \cdot \frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)\right) \]

6. Simplified 99.1%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}{{\left(\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}}}\right)\right) \]
   Proof

   [Start] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)}^{2}} \cdot \frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}\right)\right) \]
   associate-*l/ [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)}^{2}}}\right)\right) \]
   *-lft-identity [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}}}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)}^{2}}\right)\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{\color{blue}{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)}^{2}}\right)\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}{{\left(\sqrt[3]{\color{blue}{2 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2}}\right)\right) \]

7. Final simplification 99.1%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}}}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2}\right)}^{2}}\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	59264

\[\begin{array}{l} t_0 := \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\left(1 + t_0\right)}^{2}}{t_0 + 2}\right)\right) \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	32576

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

Alternative 3
Accuracy	99.1%
Cost	7232

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

Alternative 4
Accuracy	97.9%
Cost	6464

\[\cos^{-1} -1 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))