Falkner and Boettcher, Appendix B, 1

Average Error: 0.6 → 0.6

Time: 17.2s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

↓

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

↓

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

Derivation

1. Initial program 0.6

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

2. Simplified 0.6

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

   [Start] 0.6	\[ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
   difference-of-sqr-1 [=>] 0.9	\[ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(v + 1\right) \cdot \left(v - 1\right)}}\right) \]
   *-commutative [=>] 0.9	\[ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(v - 1\right) \cdot \left(v + 1\right)}}\right) \]
   cancel-sign-sub-inv [=>] 0.9	\[ \cos^{-1} \left(\frac{\color{blue}{1 + \left(-5\right) \cdot \left(v \cdot v\right)}}{\left(v - 1\right) \cdot \left(v + 1\right)}\right) \]
   +-commutative [=>] 0.9	\[ \cos^{-1} \left(\frac{\color{blue}{\left(-5\right) \cdot \left(v \cdot v\right) + 1}}{\left(v - 1\right) \cdot \left(v + 1\right)}\right) \]
   *-commutative [=>] 0.9	\[ \cos^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\left(v - 1\right) \cdot \left(v + 1\right)}\right) \]
   fma-def [=>] 0.9	\[ \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\left(v - 1\right) \cdot \left(v + 1\right)}\right) \]
   metadata-eval [=>] 0.9	\[ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\left(v - 1\right) \cdot \left(v + 1\right)}\right) \]
   *-commutative [<=] 0.9	\[ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\left(v + 1\right) \cdot \left(v - 1\right)}}\right) \]
   difference-of-sqr-1 [<=] 0.6	\[ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{v \cdot v - 1}}\right) \]
   fma-neg [=>] 0.6	\[ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]
   metadata-eval [=>] 0.6	\[ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right) \]

3. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	7232

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

Alternative 2
Error	1.3
Cost	6848

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

Alternative 3
Error	1.4
Cost	6464

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

Reproduce

herbie shell --seed 2022354 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))