Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 15.0s

Alternatives: 7

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

\[\begin{array}{l} \\ e^{{\left({\left({\log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right)}^{3}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp
  (pow
   (pow
    (pow
     (log (acos (/ (fma -5.0 (pow v 2.0) 1.0) (fma v v -1.0))))
     0.16666666666666666)
    2.0)
   3.0)))

C

double code(double v) {
	return exp(pow(pow(pow(log(acos((fma(-5.0, pow(v, 2.0), 1.0) / fma(v, v, -1.0)))), 0.16666666666666666), 2.0), 3.0));
}

Julia

function code(v)
	return exp((((log(acos(Float64(fma(-5.0, (v ^ 2.0), 1.0) / fma(v, v, -1.0)))) ^ 0.16666666666666666) ^ 2.0) ^ 3.0))
end

Wolfram

code[v_] := N[Exp[N[Power[N[Power[N[Power[N[Log[N[ArcCos[N[(N[(-5.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. add-exp-log 99.0%

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

   2. sub-neg 99.0%

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

   3. *-commutative 99.0%

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

   4. distribute-rgt-neg-in 99.0%

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

   5. pow2 99.0%

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

   6. metadata-eval 99.0%

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

   7. fmm-def 99.0%

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

   8. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

   1. add-cube-cbrt 95.1%

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

   2. pow3 95.1%

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

   3. +-commutative 95.1%

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

   4. pow2 95.1%

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

   5. *-commutative 95.1%

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

   6. fma-define 95.1%

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

   7. pow2 95.1%

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

6. Applied egg-rr 95.1%

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

   1. add-sqr-sqrt 93.6%

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

   2. pow2 93.6%

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

   3. pow1/3 99.0%

      \[\leadsto e^{{\left({\left(\sqrt{\color{blue}{{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.3333333333333333}}}\right)}^{2}\right)}^{3}} \]

   4. sqrt-pow1 99.0%

      \[\leadsto e^{{\left({\color{blue}{\left({\log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{2}\right)}^{3}} \]

   5. metadata-eval 99.0%

      \[\leadsto e^{{\left({\left({\log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\color{blue}{0.16666666666666666}}\right)}^{2}\right)}^{3}} \]

8. Applied egg-rr 99.0%

   \[\leadsto e^{{\color{blue}{\left({\left({\log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right)}}^{3}} \]
9. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[Exp[N[(N[(3.0 * N[Log[N[ArcCos[N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. add-exp-log 99.0%

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

   2. sub-neg 99.0%

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

   3. *-commutative 99.0%

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

   4. distribute-rgt-neg-in 99.0%

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

   5. pow2 99.0%

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

   6. metadata-eval 99.0%

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

   7. fmm-def 99.0%

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

   8. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

   1. pow1 99.0%

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

   2. metadata-eval 99.0%

      \[\leadsto e^{\log \left({\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}}\right)} \]

   3. pow-pow 99.0%

      \[\leadsto e^{\log \color{blue}{\left({\left({\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333}\right)}} \]

   4. log-pow 97.5%

      \[\leadsto e^{\color{blue}{0.3333333333333333 \cdot \log \left({\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}} \]

   5. *-commutative 97.5%

      \[\leadsto e^{\color{blue}{\log \left({\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right) \cdot 0.3333333333333333}} \]

   6. +-commutative 97.5%

      \[\leadsto e^{\log \left({\cos^{-1} \left(\frac{\color{blue}{{v}^{2} \cdot -5 + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right) \cdot 0.3333333333333333} \]

   7. pow2 97.5%

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

   8. *-commutative 97.5%

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

   9. fma-define 97.5%

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

   10. pow2 97.5%

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

6. Applied egg-rr 97.5%

   \[\leadsto e^{\color{blue}{\log \left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right) \cdot 0.3333333333333333}} \]
7. Step-by-step derivation

   1. pow-to-exp 99.0%

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

   2. rem-log-exp 99.0%

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

8. Applied egg-rr 99.0%

   \[\leadsto e^{\color{blue}{\left(\log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot 3\right)} \cdot 0.3333333333333333} \]
9. Step-by-step derivation

   1. pow2 99.0%

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

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

    \[\leadsto e^{\left(3 \cdot \log \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) \cdot 0.3333333333333333} \]
12. Add Preprocessing

Alternative 3: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. add-cbrt-cube 97.5%

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

   2. pow1/3 99.0%

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

   3. pow3 99.0%

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

   4. sub-neg 99.0%

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

   5. *-commutative 99.0%

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

   6. distribute-rgt-neg-in 99.0%

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

   7. pow2 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. pow2 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in v around 0 98.3%

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

4. Final simplification 98.3%

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

Alternative 6: 97.9% 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 99.0%

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

3. Taylor expanded in v around 0 98.3%

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

4. Taylor expanded in v around 0 98.2%

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

   1. metadata-eval 98.2%

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

   2. *-un-lft-identity 98.2%

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

6. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{1 \cdot \cos^{-1} -1} \]
7. Step-by-step derivation

   1. *-lft-identity 98.2%

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

8. Simplified 98.2%

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

Alternative 7: 0.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -5 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -5.0))

C

double code(double v) {
	return acos(-5.0);
}

Fortran

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

Java

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

Python

def code(v):
	return math.acos(-5.0)

Julia

function code(v)
	return acos(-5.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-5.0);
end

Wolfram

code[v_] := N[ArcCos[-5.0], $MachinePrecision]

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in v around inf 0.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))