Falkner and Boettcher, Appendix B, 1

Average Error: 0.6 → 0.8

Time: 14.8s

Precision: binary64

Cost: 13440

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (acos (+ (* 4.0 (+ (* v v) (pow v 4.0))) -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(((4.0 * ((v * v) + pow(v, 4.0))) + -1.0));
}

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) + (v ** 4.0d0))) + (-1.0d0)))
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.acos(((4.0 * ((v * v) + Math.pow(v, 4.0))) + -1.0));
}

Python

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

↓

def code(v):
	return math.acos(((4.0 * ((v * v) + math.pow(v, 4.0))) + -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(Float64(4.0 * Float64(Float64(v * v) + (v ^ 4.0))) + -1.0))
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) + (v ^ 4.0))) + -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[(4.0 * N[(N[(v * v), $MachinePrecision] + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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. Taylor expanded in v around 0 0.8

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

3. Simplified 0.8

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

   [Start] 0.8	\[ \cos^{-1} \left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right) \]
   sub-neg [=>] 0.8	\[ \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) + \left(-1\right)\right)} \]
   distribute-lft-out [=>] 0.8	\[ \cos^{-1} \left(\color{blue}{4 \cdot \left({v}^{2} + {v}^{4}\right)} + \left(-1\right)\right) \]
   unpow2 [=>] 0.8	\[ \cos^{-1} \left(4 \cdot \left(\color{blue}{v \cdot v} + {v}^{4}\right) + \left(-1\right)\right) \]
   metadata-eval [=>] 0.8	\[ \cos^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + \color{blue}{-1}\right) \]

4. Final simplification 0.8

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

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	0.9
Cost	6848

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

Alternative 3
Error	1.4
Cost	6464

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

Reproduce

herbie shell --seed 2023237 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))