Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 37.6s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 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)));
}

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

Java

public static double code(double v) {
	return Math.acos(((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)))

Julia

function code(v)
	return acos(Float64(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

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]

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 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)));
}

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

Java

public static double code(double v) {
	return Math.acos(((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)))

Julia

function code(v)
	return acos(Float64(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

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]

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - \sqrt[3]{{v}^{6} \cdot 125}}{v \cdot v + -1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (cbrt (* (pow v 6.0) 125.0))) (+ (* v v) -1.0))))

C

double code(double v) {
	return acos(((1.0 - cbrt((pow(v, 6.0) * 125.0))) / ((v * v) + -1.0)));
}

Java

public static double code(double v) {
	return Math.acos(((1.0 - Math.cbrt((Math.pow(v, 6.0) * 125.0))) / ((v * v) + -1.0)));
}

Julia

function code(v)
	return acos(Float64(Float64(1.0 - cbrt(Float64((v ^ 6.0) * 125.0))) / Float64(Float64(v * v) + -1.0)))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[Power[N[(N[Power[v, 6.0], $MachinePrecision] * 125.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 99.3%

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

   2. pow1/3 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{{\left(\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)\right)}^{0.3333333333333333}}}{v \cdot v - 1}\right) \]

   3. pow3 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\color{blue}{\left({\left(5 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

   4. *-commutative 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\left({\color{blue}{\left(\left(v \cdot v\right) \cdot 5\right)}}^{3}\right)}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

   5. unpow-prod-down 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\color{blue}{\left({\left(v \cdot v\right)}^{3} \cdot {5}^{3}\right)}}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

   6. pow2 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\left({\color{blue}{\left({v}^{2}\right)}}^{3} \cdot {5}^{3}\right)}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

   7. pow-pow 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\left(\color{blue}{{v}^{\left(2 \cdot 3\right)}} \cdot {5}^{3}\right)}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

   8. metadata-eval 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\left({v}^{\color{blue}{6}} \cdot {5}^{3}\right)}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - {\left({v}^{6} \cdot \color{blue}{125}\right)}^{0.3333333333333333}}{v \cdot v - 1}\right) \]

4. Applied egg-rr 99.3%

   \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{{\left({v}^{6} \cdot 125\right)}^{0.3333333333333333}}}{v \cdot v - 1}\right) \]
5. Step-by-step derivation

   1. unpow1/3 99.3%

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{\sqrt[3]{{v}^{6} \cdot 125}}}{v \cdot v - 1}\right) \]

6. Simplified 99.3%

   \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{\sqrt[3]{{v}^{6} \cdot 125}}}{v \cdot v - 1}\right) \]

7. Final simplification 99.3%

   \[\leadsto \cos^{-1} \left(\frac{1 - \sqrt[3]{{v}^{6} \cdot 125}}{v \cdot v + -1}\right) \]
8. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - ((v * v) * 5.0d0)) / ((v * v) + (-1.0d0))))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - ((v * v) * 5.0)) / ((v * v) + -1.0)));
}

Python

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

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / Float64(Float64(v * v) + -1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - ((v * v) * 5.0)) / ((v * v) + -1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-1.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(-1.0);
}

Python

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

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-1.0);
end

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0 98.1%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]

4. Final simplification 98.1%

   \[\leadsto \cos^{-1} -1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))