Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 17.3s

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: 99.2% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[Exp[N[Log[N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. sub-neg 98.9%

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

   2. +-commutative 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. metadata-eval 98.9%

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

   6. associate-*r* 98.9%

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

   7. fma-udef 98.9%

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

   8. add-exp-log 98.9%

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

   9. fma-neg 98.9%

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

   10. metadata-eval 98.9%

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

3. Applied egg-rr 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $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. Step-by-step derivation

   1. sub-neg 98.9%

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

   2. +-commutative 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. metadata-eval 98.9%

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

   6. associate-*r* 98.9%

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

   7. fma-udef 98.9%

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

3. Applied egg-rr 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 99.2% 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 98.9%

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

2. Final simplification 98.9%

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

Alternative 4: 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 98.9%

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

2. Taylor expanded in v around 0 97.7%

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

   1. add-sqr-sqrt 97.7%

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

   2. difference-of-sqr-1 97.6%

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

   3. *-commutative 97.6%

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

   4. sqrt-prod 97.6%

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

   5. pow2 97.6%

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

   6. sqrt-prod 54.4%

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

   7. add-sqr-sqrt 96.7%

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

   8. metadata-eval 96.7%

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

   9. *-commutative 96.7%

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

   10. sqrt-prod 96.7%

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

   11. pow2 96.7%

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

   12. sqrt-prod 54.4%

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

   13. add-sqr-sqrt 97.6%

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

   14. metadata-eval 97.6%

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

4. Applied egg-rr 97.6%

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

5. Final simplification 97.6%

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

Alternative 5: 98.2% 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. Taylor expanded in v around 0 96.8%

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

3. Final simplification 96.8%

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

Reproduce

herbie shell --seed 2023302 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))