Falkner and Boettcher, Appendix B, 1

Average Error: 99.1% → 99.1%

Time: 14.1s

Precision: binary64

Cost: 20288.00

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (+
  (+ (/ 1.0 (/ 1.0 (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))) 2.0)
  -2.0))

C

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

↓

double code(double v) {
	return ((1.0 / (1.0 / acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))))) + 2.0) + -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)
	return Float64(Float64(Float64(1.0 / Float64(1.0 / acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))) + 2.0) + -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_] := N[(N[(N[(1.0 / N[(1.0 / N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + -2.0), $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

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

3. Applied egg-rr 99.1

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

4. Applied egg-rr 99.1

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

5. Applied egg-rr 99.1

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

6. Final simplification 99.1

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

Alternatives

Alternative 1
Error	99.1%
Cost	13760.00

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

Alternative 2
Error	98.7%
Cost	7232.00

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

Alternative 3
Error	99.1%
Cost	7232.00

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

Alternative 4
Error	98.1%
Cost	6848.00

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

Alternative 5
Error	98.0%
Cost	6464.00

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

Reproduce

herbie shell --seed 2023097 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))