Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.7

Time: 19.9s

Precision: binary64

Cost: 26368

Math

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

↓

\[{\left({\cos^{-1} \left(\left(4 \cdot v\right) \cdot \left({v}^{3} + v\right) + -1\right)}^{3}\right)}^{0.3333333333333333} \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (pow
  (pow (acos (+ (* (* 4.0 v) (+ (pow v 3.0) v)) -1.0)) 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((((4.0 * v) * (pow(v, 3.0) + v)) + -1.0)), 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((((4.0d0 * v) * ((v ** 3.0d0) + v)) + (-1.0d0))) ** 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((((4.0 * v) * (Math.pow(v, 3.0) + v)) + -1.0)), 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((((4.0 * v) * (math.pow(v, 3.0) + v)) + -1.0)), 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(Float64(4.0 * v) * Float64((v ^ 3.0) + v)) + -1.0)) ^ 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((((4.0 * v) * ((v ^ 3.0) + v)) + -1.0)) ^ 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[(N[(4.0 * v), $MachinePrecision] * N[(N[Power[v, 3.0], $MachinePrecision] + v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Taylor expanded in v around 0 0.7

   \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)} \]

3. Simplified 0.7

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)} \]
   Proof

4. Applied egg-rr 0.7

   \[\leadsto \cos^{-1} \left(\color{blue}{\left(v \cdot v\right) \cdot \left(4 \cdot \left(v \cdot v\right) + 4\right)} - 1\right) \]

5. Applied egg-rr 0.7

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(\left(4 \cdot v\right) \cdot \left({v}^{3} + v\right) + -1\right)}^{3}\right)}^{0.3333333333333333}} \]

Alternatives

Alternative 1
Error	0.7
Cost	7232

\[\cos^{-1} \left(\left(v \cdot v\right) \cdot \left(4 \cdot \left(v \cdot v\right) + 4\right) - 1\right) \]

Alternative 2
Error	0.8
Cost	6848

\[\cos^{-1} \left(4 \cdot \left(v \cdot v\right) - 1\right) \]

Alternative 3
Error	1.2
Cost	6464

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

Reproduce

herbie shell --seed 2023010 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))