Falkner and Boettcher, Appendix B, 1

Average Error: 0.82% → 0.82%

Time: 14.6s

Precision: binary64

Cost: 20160

Math

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

↓

\[{\left({\cos^{-1} \left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right)}^{3}\right)}^{0.3333333333333333} \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (pow
  (pow (acos (/ (+ -1.0 (* (* v v) 5.0)) (- 1.0 (* v v)))) 3.0)
  0.3333333333333333))

C

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

↓

double code(double v) {
	return pow(pow(acos(((-1.0 + ((v * v) * 5.0)) / (1.0 - (v * v)))), 3.0), 0.3333333333333333);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

↓

real(8) function code(v)
    real(8), intent (in) :: v
    code = (acos((((-1.0d0) + ((v * v) * 5.0d0)) / (1.0d0 - (v * v)))) ** 3.0d0) ** 0.3333333333333333d0
end function

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.pow(Math.pow(Math.acos(((-1.0 + ((v * v) * 5.0)) / (1.0 - (v * v)))), 3.0), 0.3333333333333333);
}

Python

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

↓

def code(v):
	return math.pow(math.pow(math.acos(((-1.0 + ((v * v) * 5.0)) / (1.0 - (v * v)))), 3.0), 0.3333333333333333)

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(Float64(-1.0 + Float64(Float64(v * v) * 5.0)) / Float64(1.0 - Float64(v * v)))) ^ 3.0) ^ 0.3333333333333333
end

MATLAB

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

↓

function tmp = code(v)
	tmp = (acos(((-1.0 + ((v * v) * 5.0)) / (1.0 - (v * v)))) ^ 3.0) ^ 0.3333333333333333;
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[Power[N[Power[N[ArcCos[N[(N[(-1.0 + N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 0.82

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

2. Applied egg-rr 0.82

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

3. Applied egg-rr 0.86

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

4. Simplified 0.82

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

   [Start] 0.86	\[ \pi \cdot 0.5 - \sin^{-1} \left(\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{1}{1 - v \cdot v}\right) \]
   associate-*r/ [=>] 0.82	\[ \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(\frac{\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot 1}{1 - v \cdot v}\right)} \]
   *-rgt-identity [=>] 0.82	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\color{blue}{-1 - v \cdot \left(v \cdot -5\right)}}{1 - v \cdot v}\right) \]
   *-commutative [=>] 0.82	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{\left(v \cdot -5\right) \cdot v}}{1 - v \cdot v}\right) \]
   *-commutative [=>] 0.82	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{\left(-5 \cdot v\right)} \cdot v}{1 - v \cdot v}\right) \]
   associate-*r* [<=] 0.82	\[ \pi \cdot 0.5 - \sin^{-1} \left(\frac{-1 - \color{blue}{-5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\right) \]

5. Applied egg-rr 0.82

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Final simplification 0.82

   \[\leadsto {\left({\cos^{-1} \left(\frac{-1 + \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\right)}^{3}\right)}^{0.3333333333333333} \]

Alternatives

Alternative 1
Error	0.82%
Cost	13824

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

Alternative 2
Error	0.82%
Cost	7232

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

Alternative 3
Error	2%
Cost	6464

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

Reproduce

herbie shell --seed 2023115 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))