Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 16.6s

Alternatives: 9

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1
  (*
   2.0
   (log (sqrt (+ 1.0 (acos (/ (fma (* v v) -5.0 1.0) (fma v v -1.0)))))))))

C

double code(double v) {
	return expm1((2.0 * log(sqrt((1.0 + acos((fma((v * v), -5.0, 1.0) / fma(v, v, -1.0))))))));
}

Julia

function code(v)
	return expm1(Float64(2.0 * log(sqrt(Float64(1.0 + acos(Float64(fma(Float64(v * v), -5.0, 1.0) / fma(v, v, -1.0))))))))
end

Wolfram

code[v_] := N[(Exp[N[(2.0 * N[Log[N[Sqrt[N[(1.0 + N[ArcCos[N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. sub-neg 99.0%

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

   2. sqr-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-define 99.0%

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

   7. sqr-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

3. Simplified 99.0%

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

   1. metadata-eval 99.0%

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

   2. fmm-def 99.0%

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

   3. fma-undefine 99.0%

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

   4. metadata-eval 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. *-commutative 99.0%

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

   7. +-commutative 99.0%

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

   8. sub-neg 99.0%

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

   9. expm1-log1p-u 99.0%

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

   10. expm1-undefine 99.0%

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

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-define 99.0%

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

8. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 99.0%

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

   2. pow2 99.0%

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

10. Applied egg-rr 99.0%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{2}}\right) \]
11. Step-by-step derivation

    1. unpow2 99.0%

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

    2. add-sqr-sqrt 99.0%

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

    3. log1p-undefine 99.0%

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

    4. add-sqr-sqrt 99.0%

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

    5. log-prod 99.0%

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

12. Applied egg-rr 99.0%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) + \log \left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}\right) \]
13. Step-by-step derivation

    1. count-2 99.0%

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

14. Simplified 99.0%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{2 \cdot \log \left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}\right) \]
15. Step-by-step derivation

    1. unpow2 99.0%

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

16. Applied egg-rr 99.0%

    \[\leadsto \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
17. Add Preprocessing

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (-
  (* PI 0.5)
  (log1p (expm1 (asin (/ (- 1.0 (* (* v v) 5.0)) (fma v v -1.0)))))))

C

double code(double v) {
	return (((double) M_PI) * 0.5) - log1p(expm1(asin(((1.0 - ((v * v) * 5.0)) / fma(v, v, -1.0)))));
}

Julia

function code(v)
	return Float64(Float64(pi * 0.5) - log1p(expm1(asin(Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / fma(v, v, -1.0))))))
end

Wolfram

code[v_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Log[1 + N[(Exp[N[ArcSin[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. sub-neg 99.0%

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

   2. sqr-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-define 99.0%

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

   7. sqr-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

3. Simplified 99.0%

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

   1. acos-asin 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. fmm-def 99.0%

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

   5. fma-undefine 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-rgt-neg-in 99.0%

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

   8. *-commutative 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. div-inv 99.0%

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

   12. metadata-eval 99.0%

       \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 99.0%

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

8. Simplified 99.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
9. Step-by-step derivation

   1. log1p-expm1-u 99.0%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

10. Applied egg-rr 99.0%

    \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
11. Step-by-step derivation

    1. fma-undefine 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\color{blue}{{v}^{2} \cdot -5 + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    2. pow2 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\color{blue}{\left(v \cdot v\right)} \cdot -5 + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    3. metadata-eval 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\left(v \cdot v\right) \cdot \color{blue}{\left(-5\right)} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    4. distribute-rgt-neg-in 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\color{blue}{\left(-\left(v \cdot v\right) \cdot 5\right)} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    5. *-commutative 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\left(-\color{blue}{5 \cdot \left(v \cdot v\right)}\right) + 1}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    6. +-commutative 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    7. sub-neg 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

    8. pow2 99.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

12. Applied egg-rr 99.0%

    \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{\color{blue}{1 - 5 \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
13. Step-by-step derivation

    1. unpow2 99.0%

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

14. Applied egg-rr 99.0%

    \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

15. Final simplification 99.0%

    \[\leadsto \pi \cdot 0.5 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
16. Add Preprocessing

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (- (* PI 0.5) (asin (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0)))))

C

double code(double v) {
	return (((double) M_PI) * 0.5) - asin((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return Float64(Float64(pi * 0.5) - asin(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $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. Step-by-step derivation

   1. sub-neg 99.0%

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

   2. sqr-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-define 99.0%

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

   7. sqr-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

3. Simplified 99.0%

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

   1. acos-asin 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. fmm-def 99.0%

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

   5. fma-undefine 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-rgt-neg-in 99.0%

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

   8. *-commutative 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. div-inv 99.0%

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

   12. metadata-eval 99.0%

       \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 99.0%

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

8. Simplified 99.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. sub-neg 99.0%

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

   2. sqr-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-define 99.0%

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

   7. sqr-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

3. Simplified 99.0%

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

Alternative 5: 99.1% 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 99.0%

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

3. Final simplification 99.0%

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

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \sin^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (- (* PI 0.5) (asin (+ -1.0 (* (* v v) 4.0)))))

C

double code(double v) {
	return (((double) M_PI) * 0.5) - asin((-1.0 + ((v * v) * 4.0)));
}

Java

public static double code(double v) {
	return (Math.PI * 0.5) - Math.asin((-1.0 + ((v * v) * 4.0)));
}

Python

def code(v):
	return (math.pi * 0.5) - math.asin((-1.0 + ((v * v) * 4.0)))

Julia

function code(v)
	return Float64(Float64(pi * 0.5) - asin(Float64(-1.0 + Float64(Float64(v * v) * 4.0))))
end

MATLAB

function tmp = code(v)
	tmp = (pi * 0.5) - asin((-1.0 + ((v * v) * 4.0)));
end

Wolfram

code[v_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(-1.0 + N[(N[(v * v), $MachinePrecision] * 4.0), $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. Step-by-step derivation

   1. sub-neg 99.0%

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

   2. sqr-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-define 99.0%

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

   7. sqr-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

3. Simplified 99.0%

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

   1. acos-asin 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. fmm-def 99.0%

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

   5. fma-undefine 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-rgt-neg-in 99.0%

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

   8. *-commutative 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. div-inv 99.0%

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

   12. metadata-eval 99.0%

       \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 99.0%

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

8. Simplified 99.0%

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

9. Taylor expanded in v around 0 97.9%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
10. Step-by-step derivation

    1. unpow2 99.0%

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

11. Applied egg-rr 97.9%

    \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} - 1\right) \]

12. Final simplification 97.9%

    \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right) \]
13. Add Preprocessing

Alternative 7: 98.2% 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 97.3%

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

4. Final simplification 97.3%

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

Alternative 8: 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 99.0%

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

3. Taylor expanded in v around 0 97.3%

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

4. Taylor expanded in v around 0 97.1%

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

Alternative 9: 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. Step-by-step derivation

   1. sub-neg 99.0%

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

   2. sqr-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-define 99.0%

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

   7. sqr-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. fmm-def 99.0%

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

   10. metadata-eval 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in v around inf 0.0%

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

Reproduce

herbie shell --seed 2024176 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))