Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 15.9s

Alternatives: 4

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

\[\begin{array}{l} \\ e^{\log \left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}\right)}^{3}\right) \cdot 9} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp
  (*
   (log
    (pow
     (cbrt (cbrt (cbrt (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))))
     3.0))
   9.0)))

C

double code(double v) {
	return exp((log(pow(cbrt(cbrt(cbrt(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)))))), 3.0)) * 9.0));
}

Julia

function code(v)
	return exp(Float64(log((cbrt(cbrt(cbrt(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))))) ^ 3.0)) * 9.0))
end

Wolfram

code[v_] := N[Exp[N[(N[Log[N[Power[N[Power[N[Power[N[Power[N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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. Step-by-step derivation

   1. add-log-exp 99.5%

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

   2. add-cube-cbrt 99.5%

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

   3. log-prod 99.5%

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

3. Applied egg-rr 99.5%

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

   1. log-prod 99.5%

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

   2. count-2 99.5%

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

   3. distribute-lft1-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. fma-udef 99.5%

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

   6. metadata-eval 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. *-commutative 99.5%

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

   9. unpow2 99.5%

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

   10. +-commutative 99.5%

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

   11. sub-neg 99.5%

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

   12. metadata-eval 99.5%

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

   13. fma-neg 99.5%

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

5. Simplified 99.5%

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

   1. add-cube-cbrt 95.6%

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

   2. pow3 96.0%

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

7. Applied egg-rr 96.0%

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

   1. add-log-exp 96.0%

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

   2. *-commutative 96.0%

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

   3. rem-cube-cbrt 99.5%

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

   4. exp-to-pow 99.5%

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

   5. pow3 99.5%

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

   6. add-cube-cbrt 99.5%

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

   7. add-log-exp 99.5%

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

   8. add-cube-cbrt 97.1%

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

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9}} \]
10. Step-by-step derivation

    1. add-cube-cbrt 99.5%

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

    2. pow3 99.5%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

    \[\leadsto e^{\log \left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}\right)}^{3}\right) \cdot 9} \]

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ e^{9 \cdot \log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp
  (*
   9.0
   (log (cbrt (cbrt (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))))))

C

double code(double v) {
	return exp((9.0 * log(cbrt(cbrt(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))))))));
}

Julia

function code(v)
	return exp(Float64(9.0 * log(cbrt(cbrt(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))))))
end

Wolfram

code[v_] := N[Exp[N[(9.0 * N[Log[N[Power[N[Power[N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]], $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. Step-by-step derivation

   1. add-log-exp 99.5%

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

   2. add-cube-cbrt 99.5%

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

   3. log-prod 99.5%

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

3. Applied egg-rr 99.5%

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

   1. log-prod 99.5%

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

   2. count-2 99.5%

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

   3. distribute-lft1-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. fma-udef 99.5%

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

   6. metadata-eval 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. *-commutative 99.5%

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

   9. unpow2 99.5%

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

   10. +-commutative 99.5%

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

   11. sub-neg 99.5%

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

   12. metadata-eval 99.5%

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

   13. fma-neg 99.5%

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

5. Simplified 99.5%

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

   1. add-cube-cbrt 95.6%

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

   2. pow3 96.0%

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

7. Applied egg-rr 96.0%

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

   1. add-log-exp 96.0%

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

   2. *-commutative 96.0%

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

   3. rem-cube-cbrt 99.5%

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

   4. exp-to-pow 99.5%

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

   5. pow3 99.5%

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

   6. add-cube-cbrt 99.5%

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

   7. add-log-exp 99.5%

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

   8. add-cube-cbrt 97.1%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

    \[\leadsto e^{9 \cdot \log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)} \]

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  3.0
  (* (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))) 0.3333333333333333)))

C

double code(double v) {
	return 3.0 * (acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))) * 0.3333333333333333);
}

Julia

function code(v)
	return Float64(3.0 * Float64(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))) * 0.3333333333333333))
end

Wolfram

code[v_] := N[(3.0 * N[(N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. 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. pow3 98.0%

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

   3. sub-neg 98.0%

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

   4. +-commutative 98.0%

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

   5. *-commutative 98.0%

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

   6. distribute-rgt-neg-in 98.0%

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

   7. fma-def 98.0%

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

   8. metadata-eval 98.0%

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

   9. fma-neg 98.0%

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

   10. metadata-eval 98.0%

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

3. Applied egg-rr 98.0%

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

   1. pow1/3 98.0%

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

   2. add-sqr-sqrt 98.0%

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

   3. unpow-prod-down 97.1%

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

   4. fma-udef 97.1%

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

   5. associate-*r* 97.1%

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

   6. fma-udef 97.1%

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

   7. fma-udef 97.1%

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

   8. associate-*r* 97.1%

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

   9. fma-udef 97.1%

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

5. Applied egg-rr 97.1%

   \[\leadsto {\color{blue}{\left({\left(\sqrt{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]
6. Step-by-step derivation

   1. unpow1/3 98.0%

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

   2. unpow1/3 94.7%

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

7. Simplified 94.7%

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

   1. unpow3 95.0%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 4: 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 99.5%

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

2. Final simplification 99.5%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))