Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 30.4s

Alternatives: 4

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

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) + -1\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp (log1p (+ (acos (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0))) -1.0))))

C

double code(double v) {
	return exp(log1p((acos(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0))) + -1.0)));
}

Julia

function code(v)
	return exp(log1p(Float64(acos(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / fma(v, v, -1.0))) + -1.0)))
end

Wolfram

code[v_] := N[Exp[N[Log[1 + N[(N[ArcCos[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-exp-log 99.3%

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

   2. pow2 99.3%

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

   3. fma-neg 99.3%

      \[\leadsto e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)} \]

3. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \]
4. Step-by-step derivation

   1. log1p-expm1-u 99.3%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}} \]

   2. expm1-udef 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{e^{\log \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} - 1}\right)} \]

   3. rem-exp-log 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} - 1\right)} \]

   4. sub-neg 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot {v}^{2}\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right) - 1\right)} \]

   5. *-commutative 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \left(-\color{blue}{{v}^{2} \cdot 5}\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) - 1\right)} \]

   6. distribute-rgt-neg-in 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot \left(-5\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right) - 1\right)} \]

   7. metadata-eval 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{\mathsf{fma}\left(v, v, -1\right)}\right) - 1\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) - 1\right)}} \]

6. Final simplification 99.3%

   \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) + -1\right)} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + -1 \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (+ (+ 1.0 (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0)))) -1.0))

C

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

Julia

function code(v)
	return Float64(Float64(1.0 + acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0)))) + -1.0)
end

Wolfram

code[v_] := N[(N[(1.0 + N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. pow2 97.8%

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

   4. fma-neg 97.8%

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

   5. metadata-eval 97.8%

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

3. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{{\left(\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. pow1/2 97.8%

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

   4. sqrt-pow1 97.8%

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

   5. sub-neg 97.8%

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

   6. *-commutative 97.8%

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

   7. distribute-rgt-neg-in 97.8%

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

   8. metadata-eval 97.8%

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

   9. metadata-eval 97.8%

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

5. Applied egg-rr 97.8%

   \[\leadsto {\color{blue}{\left({\left({\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.25}\right)}^{2}\right)}}^{2} \]
6. Step-by-step derivation

   1. pow-pow 97.8%

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

   2. +-commutative 97.8%

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

   3. fma-udef 97.8%

      \[\leadsto {\left({\cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left({v}^{2}, -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\left(0.25 \cdot 2\right)}\right)}^{2} \]

   4. metadata-eval 97.8%

      \[\leadsto {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{\color{blue}{0.5}}\right)}^{2} \]

   5. pow-pow 99.3%

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

   6. expm1-log1p-u 99.3%

      \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}}^{\left(0.5 \cdot 2\right)} \]

   7. metadata-eval 99.3%

      \[\leadsto {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)}^{\color{blue}{1}} \]

   8. pow1 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   9. expm1-udef 99.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1} \]

   10. log1p-udef 99.3%

       \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} - 1 \]

   11. add-exp-log 99.3%

       \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1 \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) - 1} \]

8. Final simplification 99.3%

   \[\leadsto \left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right) + -1 \]

Alternative 3: 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]

Derivation

1. Initial program 99.3%

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

2. Final simplification 99.3%

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

Alternative 4: 97.9% 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 99.3%

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

2. Taylor expanded in v around 0 97.0%

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

3. Final simplification 97.0%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))