Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 98.8%

Time: 12.7s

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: 98.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(v)
	return Float64(Float64(pi * 0.5) - asin(fma(4.0, fma(v, v, (v ^ 4.0)), -1.0)))
end

Wolfram

code[v_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(4.0 * N[(v * v + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $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. Taylor expanded in v around 0 99.2%

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

   1. sub-neg 99.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) + \left(-1\right)\right)} \]

   2. unpow2 99.2%

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

   3. distribute-lft-out 99.2%

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

   4. metadata-eval 99.2%

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

4. Simplified 99.2%

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

   1. acos-asin 99.2%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + -1\right)} \]

   2. sub-neg 99.2%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + -1\right)\right)} \]

   3. div-inv 99.2%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + -1\right)\right) \]

   4. metadata-eval 99.2%

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

   5. fma-def 99.2%

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

   6. fma-def 99.2%

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right) \]

Alternative 2: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[ArcCos[N[(-1.0 + N[(4.0 * N[(N[Power[v, 4.0], $MachinePrecision] + N[(v * v), $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. Taylor expanded in v around 0 99.2%

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

   1. sub-neg 99.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) + \left(-1\right)\right)} \]

   2. unpow2 99.2%

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

   3. distribute-lft-out 99.2%

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

   4. metadata-eval 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $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. Final simplification 99.2%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[ArcCos[N[(-1.0 + N[(v * N[(4.0 * 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. Taylor expanded in v around 0 99.2%

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

   1. sub-neg 99.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) + \left(-1\right)\right)} \]

   2. unpow2 99.2%

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

   3. distribute-lft-out 99.2%

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

   4. metadata-eval 99.2%

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

4. Simplified 99.2%

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

5. Taylor expanded in v around 0 99.1%

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

   1. unpow2 99.1%

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

   2. associate-*r* 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

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. Taylor expanded in v around 0 98.4%

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

3. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))