Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 17.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-5, {v}^{2}, 1\right)\\ {\left(\sqrt[3]{\cos^{-1} \left(\frac{t\_0}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(t\_0 \cdot \frac{1}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (fma -5.0 (pow v 2.0) 1.0)))
   (*
    (pow (cbrt (acos (/ t_0 (fma v v -1.0)))) 2.0)
    (pow (pow (cbrt (acos (* t_0 (/ 1.0 (fma v v -1.0))))) 0.25) 4.0))))

C

double code(double v) {
	double t_0 = fma(-5.0, pow(v, 2.0), 1.0);
	return pow(cbrt(acos((t_0 / fma(v, v, -1.0)))), 2.0) * pow(pow(cbrt(acos((t_0 * (1.0 / fma(v, v, -1.0))))), 0.25), 4.0);
}

Julia

function code(v)
	t_0 = fma(-5.0, (v ^ 2.0), 1.0)
	return Float64((cbrt(acos(Float64(t_0 / fma(v, v, -1.0)))) ^ 2.0) * ((cbrt(acos(Float64(t_0 * Float64(1.0 / fma(v, v, -1.0))))) ^ 0.25) ^ 4.0))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(-5.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[Power[N[Power[N[ArcCos[N[(t$95$0 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[N[Power[N[ArcCos[N[(t$95$0 * N[(1.0 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 0.25], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. sub-neg 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. pow2 98.9%

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

   6. metadata-eval 98.9%

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

   7. fma-neg 98.9%

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

   8. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. acos-asin 98.9%

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

   2. div-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-rgt-identity 98.9%

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

   5. sub-neg 98.9%

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

   6. add-cube-cbrt 96.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{\pi \cdot 0.5} \cdot \sqrt[3]{\pi \cdot 0.5}\right) \cdot \sqrt[3]{\pi \cdot 0.5}} + \left(-\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot 1\right)\right)\right) \]

   7. fma-undefine 96.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt[3]{\pi \cdot 0.5} \cdot \sqrt[3]{\pi \cdot 0.5}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot 1\right)}\right)\right) \]

   8. expm1-log1p-u 96.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt[3]{\pi \cdot 0.5} \cdot \sqrt[3]{\pi \cdot 0.5}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot 1\right)\right)\right)\right)}\right) \]

6. Applied egg-rr 98.9%

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

8. Applied egg-rr 96.0%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{4} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4}} \]
9. Step-by-step derivation

   1. metadata-eval 96.0%

      \[\leadsto {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   2. pow-sqr 96.0%

      \[\leadsto \color{blue}{\left({\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2} \cdot {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right)} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   3. unpow2 96.0%

      \[\leadsto \left(\color{blue}{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666} \cdot {\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)} \cdot {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   4. pow-sqr 97.4%

      \[\leadsto \left(\color{blue}{{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\left(2 \cdot 0.16666666666666666\right)}} \cdot {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   5. metadata-eval 97.4%

      \[\leadsto \left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\color{blue}{0.3333333333333333}} \cdot {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   6. unpow1/3 97.4%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}^{2}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   7. unpow2 97.4%

      \[\leadsto \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666} \cdot {\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.16666666666666666}\right)}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   8. pow-sqr 98.8%

      \[\leadsto \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\left(2 \cdot 0.16666666666666666\right)}}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   9. metadata-eval 98.8%

      \[\leadsto \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\color{blue}{0.3333333333333333}}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   10. unpow1/3 98.8%

       \[\leadsto \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

   11. unpow2 98.8%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4} \]

10. Simplified 98.8%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{0.25}\right)}^{4}} \]
11. Step-by-step derivation

    1. div-inv 98.9%

       \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(-5, {v}^{2}, 1\right) \cdot \frac{1}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{0.25}\right)}^{4} \]

12. Applied egg-rr 98.9%

    \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2} \cdot {\left({\left(\sqrt[3]{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(-5, {v}^{2}, 1\right) \cdot \frac{1}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{0.25}\right)}^{4} \]
13. Add Preprocessing

Alternative 2: 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 98.9%

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

3. Final simplification 98.9%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in v around 0 98.0%

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

   1. unpow2 98.0%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 4: 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 98.9%

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

3. Taylor expanded in v around 0 98.0%

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

   1. unpow2 98.0%

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

5. Applied egg-rr 98.0%

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

6. Taylor expanded in v around 0 96.8%

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

Alternative 5: 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 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 2024130 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))