Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 31.5s

Alternatives: 3

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: 99.2% 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: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{{\pi}^{3} \cdot 0.125 - {t_0}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (+ 1.0 (* -5.0 (pow v 2.0))) (fma v v -1.0)))))
   (/
    (- (* (pow PI 3.0) 0.125) (pow t_0 3.0))
    (+ (* (* PI PI) 0.25) (* t_0 (fma PI 0.5 t_0))))))

C

double code(double v) {
	double t_0 = asin(((1.0 + (-5.0 * pow(v, 2.0))) / fma(v, v, -1.0)));
	return ((pow(((double) M_PI), 3.0) * 0.125) - pow(t_0, 3.0)) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(v)
	t_0 = asin(Float64(Float64(1.0 + Float64(-5.0 * (v ^ 2.0))) / fma(v, v, -1.0)))
	return Float64(Float64(Float64((pi ^ 3.0) * 0.125) - (t_0 ^ 3.0)) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(1.0 + N[(-5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $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. Step-by-step derivation

   1. acos-asin 98.9%

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

   2. flip3-- 98.9%

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

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
5. Step-by-step derivation

   1. div-sub 98.9%

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

6. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]

7. Final simplification 98.9%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - {\left(\sqrt[3]{{v}^{2} \cdot 5}\right)}^{3}}{-1 + v \cdot v}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (pow (cbrt (* (pow v 2.0) 5.0)) 3.0)) (+ -1.0 (* v v)))))

C

double code(double v) {
	return acos(((1.0 - pow(cbrt((pow(v, 2.0) * 5.0)), 3.0)) / (-1.0 + (v * v))));
}

Java

public static double code(double v) {
	return Math.acos(((1.0 - Math.pow(Math.cbrt((Math.pow(v, 2.0) * 5.0)), 3.0)) / (-1.0 + (v * v))));
}

Julia

function code(v)
	return acos(Float64(Float64(1.0 - (cbrt(Float64((v ^ 2.0) * 5.0)) ^ 3.0)) / Float64(-1.0 + Float64(v * v))))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[Power[N[Power[N[(N[Power[v, 2.0], $MachinePrecision] * 5.0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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. Step-by-step derivation

   1. add-cube-cbrt 98.9%

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

   2. pow3 98.9%

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

   3. pow2 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

   \[\leadsto \cos^{-1} \left(\frac{1 - {\left(\sqrt[3]{{v}^{2} \cdot 5}\right)}^{3}}{-1 + v \cdot v}\right) \]
6. Add Preprocessing

Alternative 3: 99.2% 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

Reproduce

herbie shell --seed 2024020 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))