Falkner and Boettcher, Appendix B, 1

Average Error: 0.6 → 0.6

Time: 25.4s

Precision: binary64

Cost: 56064

Math

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

↓

\[\begin{array}{l} t_0 := \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\\ t_1 := t_0 \cdot t_0\\ \left(0 - \left(-1 - \left(t_1 \cdot t_1\right) \cdot \frac{1}{{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)\right) - 1 \end{array} \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0 (acos (/ (- 1.0 (* 5.0 (* v v))) (+ (* v v) -1.0))))
        (t_1 (* t_0 t_0)))
   (-
    (-
     0.0
     (-
      -1.0
      (*
       (* t_1 t_1)
       (/
        1.0
        (pow
         (acos (/ (- 1.0 (* 5.0 (pow v 2.0))) (- (pow v 2.0) 1.0)))
         3.0)))))
    1.0)))

C

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

↓

double code(double v) {
	double t_0 = acos(((1.0 - (5.0 * (v * v))) / ((v * v) + -1.0)));
	double t_1 = t_0 * t_0;
	return (0.0 - (-1.0 - ((t_1 * t_1) * (1.0 / pow(acos(((1.0 - (5.0 * pow(v, 2.0))) / (pow(v, 2.0) - 1.0))), 3.0))))) - 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) + (-1.0d0))))
    t_1 = t_0 * t_0
    code = (0.0d0 - ((-1.0d0) - ((t_1 * t_1) * (1.0d0 / (acos(((1.0d0 - (5.0d0 * (v ** 2.0d0))) / ((v ** 2.0d0) - 1.0d0))) ** 3.0d0))))) - 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) {
	double t_0 = Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) + -1.0)));
	double t_1 = t_0 * t_0;
	return (0.0 - (-1.0 - ((t_1 * t_1) * (1.0 / Math.pow(Math.acos(((1.0 - (5.0 * Math.pow(v, 2.0))) / (Math.pow(v, 2.0) - 1.0))), 3.0))))) - 1.0;
}

Python

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

↓

def code(v):
	t_0 = math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) + -1.0)))
	t_1 = t_0 * t_0
	return (0.0 - (-1.0 - ((t_1 * t_1) * (1.0 / math.pow(math.acos(((1.0 - (5.0 * math.pow(v, 2.0))) / (math.pow(v, 2.0) - 1.0))), 3.0))))) - 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)
	t_0 = acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) + -1.0)))
	t_1 = Float64(t_0 * t_0)
	return Float64(Float64(0.0 - Float64(-1.0 - Float64(Float64(t_1 * t_1) * Float64(1.0 / (acos(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / Float64((v ^ 2.0) - 1.0))) ^ 3.0))))) - 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)
	t_0 = acos(((1.0 - (5.0 * (v * v))) / ((v * v) + -1.0)));
	t_1 = t_0 * t_0;
	tmp = (0.0 - (-1.0 - ((t_1 * t_1) * (1.0 / (acos(((1.0 - (5.0 * (v ^ 2.0))) / ((v ^ 2.0) - 1.0))) ^ 3.0))))) - 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_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, N[(N[(0.0 - N[(-1.0 - N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(1.0 / N[Power[N[ArcCos[N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[v, 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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. Applied egg-rr 0.6

   \[\leadsto \color{blue}{\left(0 - \left(-1 - \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right)\right) - 1} \]

3. Applied egg-rr 0.6

   \[\leadsto \left(0 - \left(-1 - \color{blue}{\left(\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right) \cdot \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right)\right) \cdot \frac{\frac{1}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)}}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)}}\right)\right) - 1 \]

4. Taylor expanded in v around 0 0.6

   \[\leadsto \left(0 - \left(-1 - \left(\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right) \cdot \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right)\right) \cdot \color{blue}{\frac{1}{{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}}\right)\right) - 1 \]

5. Final simplification 0.6

   \[\leadsto \left(0 - \left(-1 - \left(\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right) \cdot \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right)\right) \cdot \frac{1}{{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)\right) - 1 \]

Alternatives

Alternative 1
Error	0.6
Cost	7616

\[\left(0 - \left(-1 - \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v + -1}\right)\right)\right) - 1 \]

Alternative 2
Error	0.6
Cost	7232

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

Alternative 3
Error	1.3
Cost	6464

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

Reproduce

herbie shell --seed 2023077 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))