Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 35.2s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{{\pi}^{3} \cdot 0.125 - {t\_0}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t\_0 \cdot \left(\sin^{-1} \left(\frac{1 - {v}^{2} \cdot 5}{{v}^{2} + -1}\right) + \pi \cdot 0.5\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(v)
	t_0 = asin(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / 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 * Float64(asin(Float64(Float64(1.0 - Float64((v ^ 2.0) * 5.0)) / Float64((v ^ 2.0) + -1.0))) + Float64(pi * 0.5)))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $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[(N[ArcSin[N[(N[(1.0 - N[(N[Power[v, 2.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[v, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(Pi * 0.5), $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. acos-asin 99.2%

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

   2. flip3-- 99.2%

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

4. Applied egg-rr 99.2%

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

   1. cube-prod 99.2%

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

   2. metadata-eval 99.2%

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

   3. cancel-sign-sub-inv 99.2%

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

   4. metadata-eval 99.2%

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

   5. *-commutative 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in v around inf 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 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 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 3: 98.1% 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 98.5%

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

4. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024039 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))