Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 30.6s

Alternatives: 7

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, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (pow
  (pow (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))) 3.0)
  0.3333333333333333))

C

double code(double v) {
	return pow(pow(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))), 3.0), 0.3333333333333333);
}

Julia

function code(v)
	return (acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))) ^ 3.0) ^ 0.3333333333333333
end

Wolfram

code[v_] := N[Power[N[Power[N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. sub-neg 98.8%

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

   2. +-commutative 98.8%

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

   3. *-commutative 98.8%

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

   4. distribute-rgt-neg-in 98.8%

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

   5. metadata-eval 98.8%

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

   6. associate-*r* 98.8%

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

   7. fma-udef 98.8%

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

   8. fma-neg 98.8%

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

   9. metadata-eval 98.8%

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

   10. add-cbrt-cube 97.3%

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

   11. pow1/3 98.8%

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

   12. pow3 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (- (* PI 0.5) (asin (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))

C

double code(double v) {
	return (((double) M_PI) * 0.5) - asin((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return Float64(Float64(pi * 0.5) - asin(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. acos-asin 98.8%

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

   2. sub-neg 98.8%

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

   3. div-inv 98.8%

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

   4. metadata-eval 98.8%

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

   5. sub-neg 98.8%

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

   6. +-commutative 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-neg-in 98.8%

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

   9. metadata-eval 98.8%

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

   10. associate-*r* 98.8%

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

   11. fma-udef 98.8%

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

   12. fma-neg 98.8%

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

   13. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

   1. sub-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. sub-neg 98.8%

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

   2. sqr-neg 98.8%

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

   3. +-commutative 98.8%

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

   4. associate-*r* 98.8%

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

   5. distribute-rgt-neg-in 98.8%

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

   6. remove-double-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. fma-def 98.8%

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

   9. distribute-rgt-neg-out 98.8%

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

   10. *-commutative 98.8%

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

   11. distribute-rgt-neg-in 98.8%

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

   12. metadata-eval 98.8%

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

   13. fma-neg 98.8%

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

   14. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 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 98.8%

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

2. Final simplification 98.8%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double v) {
	return acos(((1.0 + (v * 2.0)) * (-1.0 + (v * 2.0))));
}

Fortran

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

Java

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

Python

def code(v):
	return math.acos(((1.0 + (v * 2.0)) * (-1.0 + (v * 2.0))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in v around 0 98.3%

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

   1. add-sqr-sqrt 98.3%

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

   2. difference-of-sqr-1 98.2%

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

   3. *-commutative 98.2%

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

   4. sqrt-prod 98.2%

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

   5. unpow2 98.2%

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

   6. sqrt-prod 44.7%

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

   7. add-sqr-sqrt 97.5%

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

   8. metadata-eval 97.5%

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

   9. *-commutative 97.5%

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

   10. sqrt-prod 97.5%

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

   11. unpow2 97.5%

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

   12. sqrt-prod 44.7%

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

   13. add-sqr-sqrt 98.2%

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

   14. metadata-eval 98.2%

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

4. Applied egg-rr 98.2%

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

5. Final simplification 98.2%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. difference-of-sqr-1 97.0%

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

   3. times-frac 97.0%

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

   4. +-commutative 97.0%

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

   5. sub-neg 97.0%

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

   6. +-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. distribute-rgt-neg-in 97.0%

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

   9. metadata-eval 97.0%

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

   10. associate-*r* 97.0%

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

   11. fma-udef 97.0%

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

   12. sub-neg 97.0%

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

   13. metadata-eval 97.0%

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

3. Applied egg-rr 97.0%

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

4. Taylor expanded in v around 0 97.8%

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

   1. sub-neg 97.8%

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

   2. metadata-eval 97.8%

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

   3. +-commutative 97.8%

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

   4. neg-mul-1 97.8%

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

   5. unsub-neg 97.8%

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

6. Simplified 97.8%

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

7. Final simplification 97.8%

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

Alternative 7: 98.0% 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 98.8%

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

2. Taylor expanded in v around 0 97.8%

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

3. Final simplification 97.8%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))