Falkner and Boettcher, Appendix B, 1

Average Error: 99.1% → 99.1%

Time: 18.0s

Precision: binary64

Cost: 7488.00

Math

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

↓

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

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (acos (/ (+ 1.0 (+ 1.0 (- -1.0 (* 5.0 (* v v))))) (+ (* v v) -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(((1.0 + (1.0 + (-1.0 - (5.0 * (v * v))))) / ((v * v) + -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(((1.0d0 + (1.0d0 + ((-1.0d0) - (5.0d0 * (v * v))))) / ((v * v) + (-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(((1.0 + (1.0 + (-1.0 - (5.0 * (v * v))))) / ((v * v) + -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(((1.0 + (1.0 + (-1.0 - (5.0 * (v * v))))) / ((v * v) + -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(1.0 + Float64(1.0 + Float64(-1.0 - Float64(5.0 * Float64(v * v))))) / Float64(Float64(v * v) + -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(((1.0 + (1.0 + (-1.0 - (5.0 * (v * v))))) / ((v * v) + -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[(1.0 + N[(1.0 + N[(-1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.1

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

2. Applied egg-rr 99.1

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

   [Start] 99.1	\[ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
   expm1-log1p-u [=>] 99.1	\[ \cos^{-1} \left(\frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(5 \cdot \left(v \cdot v\right)\right)\right)}}{v \cdot v - 1}\right) \]
   expm1-udef [=>] 99.1	\[ \cos^{-1} \left(\frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(5 \cdot \left(v \cdot v\right)\right)} - 1\right)}}{v \cdot v - 1}\right) \]
   log1p-udef [=>] 99.1	\[ \cos^{-1} \left(\frac{1 - \left(e^{\color{blue}{\log \left(1 + 5 \cdot \left(v \cdot v\right)\right)}} - 1\right)}{v \cdot v - 1}\right) \]
   add-exp-log [<=] 99.1	\[ \cos^{-1} \left(\frac{1 - \left(\color{blue}{\left(1 + 5 \cdot \left(v \cdot v\right)\right)} - 1\right)}{v \cdot v - 1}\right) \]

3. Final simplification 99.1

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

Alternatives

Alternative 1
Error	99.1%
Cost	7232.00

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

Alternative 2
Error	98.0%
Cost	6848.00

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

Alternative 3
Error	97.8%
Cost	6464.00

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

Reproduce

herbie shell --seed 2023121 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))