Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.6

Time: 9.3s

Precision: binary64

Cost: 7360

Math

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

↓

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

FPCore

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

↓

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

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))) - 2.0d0) / (1.0d0 - (v * v))))
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((((1.0 + (v * (v * 5.0))) - 2.0) / (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.acos((((1.0 + (v * (v * 5.0))) - 2.0) / (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 acos(Float64(Float64(Float64(1.0 + Float64(v * Float64(v * 5.0))) - 2.0) / 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 = acos((((1.0 + (v * (v * 5.0))) - 2.0) / (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[ArcCos[N[(N[(N[(1.0 + N[(v * N[(v * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Applied egg-rr 0.6

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

3. Applied egg-rr 0.6

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

4. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.5
Cost	7232

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

Alternative 2
Error	1.2
Cost	6848

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

Alternative 3
Error	1.3
Cost	6464

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

Reproduce

herbie shell --seed 2023096 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))