Falkner and Boettcher, Appendix B, 1

Average Accuracy: 99.2% → 99.2%

Time: 13.0s

Precision: binary64

Cost: 13824

Math

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

↓

\[\pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right) \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (- (* PI 0.5) (asin (/ (+ -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 (((double) M_PI) * 0.5) - asin(((-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.PI * 0.5) - Math.asin(((-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.pi * 0.5) - math.asin(((-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 Float64(Float64(pi * 0.5) - asin(Float64(Float64(-1.0 + Float64(Float64(v * v) * 5.0)) / Float64(1.0 - Float64(v * v)))))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

↓

function tmp = code(v)
	tmp = (pi * 0.5) - asin(((-1.0 + ((v * v) * 5.0)) / (1.0 - (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[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(-1.0 + N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

3. Applied egg-rr 99.2%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{1 - v \cdot v}\right)} \]

4. Simplified 99.2%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right)} \]
   Proof

   [Start] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{1 - v \cdot v}\right) \]
   associate-*r/ [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\frac{\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot 1}{1 - v \cdot v}\right)} \]
   *-rgt-identity [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\color{blue}{-1 - v \cdot \left(v \cdot -5\right)}}{1 - v \cdot v}\right) \]
   associate-*r* [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{\left(v \cdot v\right) \cdot -5}}{1 - v \cdot v}\right) \]
   unpow2 [<=] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{{v}^{2}} \cdot -5}{1 - v \cdot v}\right) \]
   *-commutative [<=] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{-5 \cdot {v}^{2}}}{1 - v \cdot v}\right) \]
   cancel-sign-sub-inv [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\color{blue}{-1 + \left(--5\right) \cdot {v}^{2}}}{1 - v \cdot v}\right) \]
   metadata-eval [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + \color{blue}{5} \cdot {v}^{2}}{1 - v \cdot v}\right) \]
   unpow2 [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + 5 \cdot \color{blue}{\left(v \cdot v\right)}}{1 - v \cdot v}\right) \]
   *-commutative [=>] 99.2	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + \color{blue}{\left(v \cdot v\right) \cdot 5}}{1 - v \cdot v}\right) \]

5. Final simplification 99.2%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right) \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	7232

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

Alternative 2
Accuracy	97.9%
Cost	6464

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

Reproduce

herbie shell --seed 2023129 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))