Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 18.5s

Alternatives: 6

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-cbrt-cube 97.7%

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

   2. pow1/3 99.2%

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

   3. pow3 99.2%

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

   4. sub-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. pow2 99.2%

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

   8. metadata-eval 99.2%

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

   9. fma-neg 99.2%

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

   10. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow1/3 97.7%

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

   2. rem-cbrt-cube 99.2%

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

   3. acos-asin 99.2%

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

   4. div-inv 99.2%

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

   5. metadata-eval 99.2%

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

   6. flip-- 1.7%

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

   7. clear-num 1.7%

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

   8. clear-num 1.7%

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

6. Applied egg-rr 99.2%

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

   1. fma-undefine 99.2%

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

   2. *-commutative 99.2%

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

   3. +-commutative 99.2%

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

   4. *-commutative 99.2%

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

   5. metadata-eval 99.2%

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

   6. cancel-sign-sub-inv 99.2%

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

   7. sub-div 99.2%

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

   8. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. fma-undefine 99.2%

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

   2. difference-of-sqr--1 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 1.0d0 / (1.0d0 / acos(((1.0d0 - ((v ** 2.0d0) * 5.0d0)) / ((-1.0d0) + (v ** 2.0d0)))))
end function

Java

public static double code(double v) {
	return 1.0 / (1.0 / Math.acos(((1.0 - (Math.pow(v, 2.0) * 5.0)) / (-1.0 + Math.pow(v, 2.0)))));
}

Python

def code(v):
	return 1.0 / (1.0 / math.acos(((1.0 - (math.pow(v, 2.0) * 5.0)) / (-1.0 + math.pow(v, 2.0)))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = 1.0 / (1.0 / acos(((1.0 - ((v ^ 2.0) * 5.0)) / (-1.0 + (v ^ 2.0)))));
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-cbrt-cube 97.7%

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

   2. pow1/3 99.2%

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

   3. pow3 99.2%

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

   4. sub-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. pow2 99.2%

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

   8. metadata-eval 99.2%

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

   9. fma-neg 99.2%

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

   10. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow1/3 97.7%

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

   2. rem-cbrt-cube 99.2%

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

   3. acos-asin 99.2%

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

   4. div-inv 99.2%

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

   5. metadata-eval 99.2%

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

   6. flip-- 1.7%

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

   7. clear-num 1.7%

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

   8. clear-num 1.7%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in v around inf 99.2%

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

8. Final simplification 99.2%

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

Alternative 3: 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 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0 98.2%

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

Alternative 5: 97.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

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

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0 98.2%

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

4. Taylor expanded in v around 0 98.1%

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

   1. metadata-eval 98.1%

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

   2. *-un-lft-identity 98.1%

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

6. Applied egg-rr 98.1%

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

   1. *-lft-identity 98.1%

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

8. Simplified 98.1%

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

Alternative 6: 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 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 2024139 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))