Falkner and Boettcher, Appendix B, 1

Average Error: 0.78% → 0.78%

Time: 14.9s

Precision: binary64

Cost: 20032

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (expm1 (log1p (acos (/ (+ -1.0 (* v (* v 5.0))) (- 1.0 (* v v)))))))

C

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

↓

double code(double v) {
	return expm1(log1p(acos(((-1.0 + (v * (v * 5.0))) / (1.0 - (v * v))))));
}

Java

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

↓

public static double code(double v) {
	return Math.expm1(Math.log1p(Math.acos(((-1.0 + (v * (v * 5.0))) / (1.0 - (v * v))))));
}

Python

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

↓

def code(v):
	return math.expm1(math.log1p(math.acos(((-1.0 + (v * (v * 5.0))) / (1.0 - (v * v))))))

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)
	return expm1(log1p(acos(Float64(Float64(-1.0 + Float64(v * Float64(v * 5.0))) / Float64(1.0 - Float64(v * v))))))
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_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(-1.0 + N[(v * N[(v * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 0.78

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

2. Applied egg-rr 0.78

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

3. Applied egg-rr 0.78

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

4. Simplified 0.78

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

   [Start] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(-\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{1 - v \cdot v}\right)\right)\right) \]
   distribute-neg-frac [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\frac{-\mathsf{fma}\left(v, v \cdot -5, 1\right)}{1 - v \cdot v}\right)}\right)\right) \]
   neg-sub0 [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{0 - \mathsf{fma}\left(v, v \cdot -5, 1\right)}}{1 - v \cdot v}\right)\right)\right) \]
   fma-udef [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{0 - \color{blue}{\left(v \cdot \left(v \cdot -5\right) + 1\right)}}{1 - v \cdot v}\right)\right)\right) \]
   +-commutative [<=] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{0 - \color{blue}{\left(1 + v \cdot \left(v \cdot -5\right)\right)}}{1 - v \cdot v}\right)\right)\right) \]
   associate-*r* [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{0 - \left(1 + \color{blue}{\left(v \cdot v\right) \cdot -5}\right)}{1 - v \cdot v}\right)\right)\right) \]
   *-commutative [<=] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{0 - \left(1 + \color{blue}{-5 \cdot \left(v \cdot v\right)}\right)}{1 - v \cdot v}\right)\right)\right) \]
   metadata-eval [<=] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{0 - \left(1 + \color{blue}{\left(-5\right)} \cdot \left(v \cdot v\right)\right)}{1 - v \cdot v}\right)\right)\right) \]
   cancel-sign-sub-inv [<=] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{0 - \color{blue}{\left(1 - 5 \cdot \left(v \cdot v\right)\right)}}{1 - v \cdot v}\right)\right)\right) \]
   associate--r- [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{\left(0 - 1\right) + 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\right)\right)\right) \]
   metadata-eval [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{-1} + 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right)\right)\right) \]
   *-commutative [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{-1 + \color{blue}{\left(v \cdot v\right) \cdot 5}}{1 - v \cdot v}\right)\right)\right) \]
   associate-*l* [=>] 0.78	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{-1 + \color{blue}{v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v}\right)\right)\right) \]

5. Final simplification 0.78

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

Alternatives

Alternative 1
Error	0.78%
Cost	13760

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

Alternative 2
Error	0.78%
Cost	7232

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

Alternative 3
Error	1.91%
Cost	6720

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

Alternative 4
Error	1.95%
Cost	6464

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

Reproduce

herbie shell --seed 2023102 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))