Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 36.6s

Alternatives: 5

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

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

FPCore

(FPCore (v)
 :precision binary64
 (*
  (pow
   (cbrt
    (acos (/ (- 1.0 (+ (+ 1.0 (* (pow v 2.0) 5.0)) -1.0)) (fma v v -1.0))))
   2.0)
  (/
   1.0
   (cbrt (/ 1.0 (acos (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0))))))))

C

double code(double v) {
	return pow(cbrt(acos(((1.0 - ((1.0 + (pow(v, 2.0) * 5.0)) + -1.0)) / fma(v, v, -1.0)))), 2.0) * (1.0 / cbrt((1.0 / acos(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0))))));
}

Julia

function code(v)
	return Float64((cbrt(acos(Float64(Float64(1.0 - Float64(Float64(1.0 + Float64((v ^ 2.0) * 5.0)) + -1.0)) / fma(v, v, -1.0)))) ^ 2.0) * Float64(1.0 / cbrt(Float64(1.0 / acos(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / fma(v, v, -1.0)))))))
end

Wolfram

code[v_] := N[(N[Power[N[Power[N[ArcCos[N[(N[(1.0 - N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / N[Power[N[(1.0 / N[ArcCos[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. add-cube-cbrt 97.1%

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

   2. pow2 97.1%

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

   3. pow2 97.1%

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

   4. fma-neg 97.1%

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

   5. metadata-eval 97.1%

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

   6. pow2 97.1%

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

   7. fma-neg 97.1%

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

   8. metadata-eval 97.1%

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

4. Applied egg-rr 97.1%

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

   1. expm1-log1p-u 97.1%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(5 \cdot {v}^{2}\right)\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

   2. expm1-undefine 97.1%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(5 \cdot {v}^{2}\right)} - 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

   3. log1p-undefine 97.1%

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

   4. add-exp-log 97.1%

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

   5. *-commutative 97.1%

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

6. Applied egg-rr 97.1%

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

   1. acos-asin 97.1%

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

   2. div-inv 97.1%

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

   3. metadata-eval 97.1%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(\left(1 + {v}^{2} \cdot 5\right) - 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \sqrt[3]{\pi \cdot \color{blue}{0.5} - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

   4. flip-- 0.6%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(\left(1 + {v}^{2} \cdot 5\right) - 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}{\pi \cdot 0.5 + \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}} \]

   5. clear-num 0.6%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(\left(1 + {v}^{2} \cdot 5\right) - 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\pi \cdot 0.5 + \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}} \]

   6. cbrt-div 0.6%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(\left(1 + {v}^{2} \cdot 5\right) - 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\pi \cdot 0.5 + \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}} \]

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - \left(\left(1 + {v}^{2} \cdot 5\right) + -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\frac{1}{\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}} \]
10. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right) \cdot 9} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp (* (log (cbrt (cbrt (acos (fma 4.0 (pow v 2.0) -1.0))))) 9.0)))

C

double code(double v) {
	return exp((log(cbrt(cbrt(acos(fma(4.0, pow(v, 2.0), -1.0))))) * 9.0));
}

Julia

function code(v)
	return exp(Float64(log(cbrt(cbrt(acos(fma(4.0, (v ^ 2.0), -1.0))))) * 9.0))
end

Wolfram

code[v_] := N[Exp[N[(N[Log[N[Power[N[Power[N[ArcCos[N[(4.0 * N[Power[v, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 9.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0 99.5%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
4. Step-by-step derivation

   1. add-cube-cbrt 97.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)} \cdot \sqrt[3]{\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)}} \]

   2. pow3 98.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)}\right)}^{3}} \]

   3. fma-neg 98.0%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right)}^{3} \]

   4. metadata-eval 98.0%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, \color{blue}{-1}\right)\right)}\right)}^{3} \]

5. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}\right)}^{3}} \]

6. Taylor expanded in v around 0 98.0%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)}\right)}}^{3} \]
7. Step-by-step derivation

   1. fma-neg 98.0%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right)}^{3} \]

   2. metadata-eval 98.0%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, \color{blue}{-1}\right)\right)}\right)}^{3} \]

   3. add-cube-cbrt 97.1%

      \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right)}}^{3} \]

   4. unpow3 97.1%

      \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right)}^{3}\right)}}^{3} \]

   5. pow-pow 98.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right)}^{\left(3 \cdot 3\right)}} \]

   6. pow-to-exp 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right) \cdot \left(3 \cdot 3\right)}} \]

   7. metadata-eval 99.5%

      \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right) \cdot \color{blue}{9}} \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right) \cdot 9}} \]

9. Final simplification 99.5%

   \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(4, {v}^{2}, -1\right)\right)}}\right) \cdot 9} \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left({v}^{2} \cdot 4 + -1\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (+ (* (pow v 2.0) 4.0) -1.0)))

C

double code(double v) {
	return acos(((pow(v, 2.0) * 4.0) + -1.0));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((((v ** 2.0d0) * 4.0d0) + (-1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((Math.pow(v, 2.0) * 4.0) + -1.0));
}

Python

def code(v):
	return math.acos(((math.pow(v, 2.0) * 4.0) + -1.0))

Julia

function code(v)
	return acos(Float64(Float64((v ^ 2.0) * 4.0) + -1.0))
end

MATLAB

function tmp = code(v)
	tmp = acos((((v ^ 2.0) * 4.0) + -1.0));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0 99.5%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]

4. Final simplification 99.5%

   \[\leadsto \cos^{-1} \left({v}^{2} \cdot 4 + -1\right) \]
5. Add Preprocessing

Alternative 4: 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]

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 5: 97.8% 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 99.5%

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

3. Taylor expanded in v around 0 99.3%

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

4. Final simplification 99.3%

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

Reproduce

herbie shell --seed 2024074 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))