Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 32.1s

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: 99.1% 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: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ e^{{\left({\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp
  (pow
   (pow
    (cbrt (cbrt (log (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0))))))
    3.0)
   3.0)))

C

double code(double v) {
	return exp(pow(pow(cbrt(cbrt(log(acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))), 3.0), 3.0));
}

Julia

function code(v)
	return exp(((cbrt(cbrt(log(acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))) ^ 3.0) ^ 3.0))
end

Wolfram

code[v_] := N[Exp[N[Power[N[Power[N[Power[N[Power[N[Log[N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 3.0], $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-exp-log 99.3%

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

   2. pow2 99.3%

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

   3. fma-neg 99.3%

      \[\leadsto e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)} \]

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \]
5. Step-by-step derivation

   1. add-cube-cbrt 95.5%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) \cdot \sqrt[3]{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}} \]

   2. pow3 95.5%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}}} \]

   3. sub-neg 95.5%

      \[\leadsto e^{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot {v}^{2}\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}} \]

   4. *-commutative 95.5%

      \[\leadsto e^{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{1 + \left(-\color{blue}{{v}^{2} \cdot 5}\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}} \]

   5. distribute-rgt-neg-in 95.5%

      \[\leadsto e^{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot \left(-5\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}} \]

   6. metadata-eval 95.5%

      \[\leadsto e^{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}} \]

6. Applied egg-rr 95.5%

   \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.3%

      \[\leadsto e^{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot \sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}}^{3}} \]

   2. pow3 99.3%

      \[\leadsto e^{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}}^{3}} \]

   3. acos-asin 99.3%

      \[\leadsto e^{{\left({\left(\sqrt[3]{\sqrt[3]{\log \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}}\right)}^{3}\right)}^{3}} \]

   4. acos-asin 99.3%

      \[\leadsto e^{{\left({\left(\sqrt[3]{\sqrt[3]{\log \color{blue}{\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}\right)}^{3}\right)}^{3}} \]

   5. +-commutative 99.3%

      \[\leadsto e^{{\left({\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\color{blue}{{v}^{2} \cdot -5 + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3}} \]

   6. fma-define 99.3%

      \[\leadsto e^{{\left({\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left({v}^{2}, -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3}} \]

8. Applied egg-rr 99.3%

   \[\leadsto e^{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}}^{3}} \]
9. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 98.0% 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 97.9%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))