?

Average Error: 0.5 → 0.6
Time: 13.9s
Precision: binary64
Cost: 7360

?

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\cos^{-1} \left(\frac{1}{1 - v \cdot v} \cdot \left(-1 + \left(v \cdot v\right) \cdot 5\right)\right) \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (acos (* (/ 1.0 (- 1.0 (* v v))) (+ -1.0 (* (* v v) 5.0)))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return acos(((1.0 / (1.0 - (v * v))) * (-1.0 + ((v * v) * 5.0))));
}

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 / (1.0d0 - (v * v))) * ((-1.0d0) + ((v * v) * 5.0d0))))
end function

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 / (1.0 - (v * v))) * (-1.0 + ((v * v) * 5.0))));
}

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

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(1.0 - Float64(v * v))) * Float64(-1.0 + Float64(Float64(v * v) * 5.0))))
end

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 / (1.0 - (v * v))) * (-1.0 + ((v * v) * 5.0))));
end

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[(1.0 / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-rr0.6

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

  3. Final simplification0.6

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

Alternatives

Alternative 1
Error0.5
Cost7232
\[\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \]
Alternative 2
Error1.3
Cost6464
\[\cos^{-1} -1 \]

Error

Reproduce?

herbie shell --seed 2023018 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))