Falkner and Boettcher, Appendix B, 1

Average Error: 0.85% → 0.86%

Time: 15.9s

Precision: binary64

Cost: 78080

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0 (sqrt (log1p (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))))
   (+ -1.0 (pow (exp t_0) t_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 = sqrt(log1p(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)))));
	return -1.0 + pow(exp(t_0), t_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 = sqrt(log1p(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))))
	return Float64(-1.0 + (exp(t_0) ^ t_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[Sqrt[N[Log[1 + N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(-1.0 + N[Power[N[Exp[t$95$0], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.85

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

2. Applied egg-rr 0.85

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

3. Applied egg-rr 0.86

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

4. Final simplification 0.86

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

Alternatives

Alternative 1
Error	0.85%
Cost	26432

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

Alternative 2
Error	0.87%
Cost	14016

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

Alternative 3
Error	1.36%
Cost	7232

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

Alternative 4
Error	0.85%
Cost	7232

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

Alternative 5
Error	1.97%
Cost	6848

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

Alternative 6
Error	2.1%
Cost	6464

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

Reproduce

herbie shell --seed 2023089 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))