Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 98.8%

Time: 29.7s

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: 99.2% 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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)}\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1
  (pow
   (sqrt (log1p (acos (+ (* 4.0 (+ (pow v 2.0) (pow v 4.0))) -1.0))))
   2.0)))

C

double code(double v) {
	return expm1(pow(sqrt(log1p(acos(((4.0 * (pow(v, 2.0) + pow(v, 4.0))) + -1.0)))), 2.0));
}

Java

public static double code(double v) {
	return Math.expm1(Math.pow(Math.sqrt(Math.log1p(Math.acos(((4.0 * (Math.pow(v, 2.0) + Math.pow(v, 4.0))) + -1.0)))), 2.0));
}

Python

def code(v):
	return math.expm1(math.pow(math.sqrt(math.log1p(math.acos(((4.0 * (math.pow(v, 2.0) + math.pow(v, 4.0))) + -1.0)))), 2.0))

Julia

function code(v)
	return expm1((sqrt(log1p(acos(Float64(Float64(4.0 * Float64((v ^ 2.0) + (v ^ 4.0))) + -1.0)))) ^ 2.0))
end

Wolfram

code[v_] := N[(Exp[N[Power[N[Sqrt[N[Log[1 + N[ArcCos[N[(N[(4.0 * N[(N[Power[v, 2.0], $MachinePrecision] + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. pow2 99.0%

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

   3. fma-neg 99.0%

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

   4. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\right)\right) \]

7. Simplified 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)}\right)\right) \]
8. Step-by-step derivation

   1. add-sqr-sqrt 99.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)}}\right) \]

   2. pow2 99.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)}\right)}^{2}}\right) \]

   3. fma-define 99.0%

      \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, {v}^{2} + {v}^{4}, -1\right)\right)}\right)}\right)}^{2}\right) \]

   4. unpow2 99.0%

      \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \color{blue}{v \cdot v} + {v}^{4}, -1\right)\right)\right)}\right)}^{2}\right) \]

   5. fma-define 99.0%

      \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \color{blue}{\mathsf{fma}\left(v, v, {v}^{4}\right)}, -1\right)\right)\right)}\right)}^{2}\right) \]

9. Applied egg-rr 99.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)}\right)}^{2}}\right) \]

10. Taylor expanded in v around 0 99.0%

    \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\color{blue}{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\right)}\right)}^{2}\right) \]

11. Final simplification 99.0%

    \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)}\right)}^{2}\right) \]
12. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\log \left(1 + \cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1 (log (+ 1.0 (acos (+ (* 4.0 (+ (pow v 2.0) (pow v 4.0))) -1.0))))))

C

double code(double v) {
	return expm1(log((1.0 + acos(((4.0 * (pow(v, 2.0) + pow(v, 4.0))) + -1.0)))));
}

Java

public static double code(double v) {
	return Math.expm1(Math.log((1.0 + Math.acos(((4.0 * (Math.pow(v, 2.0) + Math.pow(v, 4.0))) + -1.0)))));
}

Python

def code(v):
	return math.expm1(math.log((1.0 + math.acos(((4.0 * (math.pow(v, 2.0) + math.pow(v, 4.0))) + -1.0)))))

Julia

function code(v)
	return expm1(log(Float64(1.0 + acos(Float64(Float64(4.0 * Float64((v ^ 2.0) + (v ^ 4.0))) + -1.0)))))
end

Wolfram

code[v_] := N[(Exp[N[Log[N[(1.0 + N[ArcCos[N[(N[(4.0 * N[(N[Power[v, 2.0], $MachinePrecision] + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. pow2 99.0%

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

   3. fma-neg 99.0%

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

   4. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\right)\right) \]

7. Simplified 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)}\right)\right) \]

8. Taylor expanded in v around 0 99.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)\right)}\right) \]

9. Final simplification 99.0%

   \[\leadsto \mathsf{expm1}\left(\log \left(1 + \cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \mathsf{fma}\left(v, v, {v}^{4}\right) + -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1 (log1p (acos (+ (* 4.0 (fma v v (pow v 4.0))) -1.0)))))

C

double code(double v) {
	return expm1(log1p(acos(((4.0 * fma(v, v, pow(v, 4.0))) + -1.0))));
}

Julia

function code(v)
	return expm1(log1p(acos(Float64(Float64(4.0 * fma(v, v, (v ^ 4.0))) + -1.0))))
end

Wolfram

code[v_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(4.0 * N[(v * v + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. pow2 99.0%

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

   3. fma-neg 99.0%

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

   4. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\right)\right) \]

7. Simplified 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) + -1\right)}\right)\right) \]

8. Taylor expanded in v around 0 99.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\color{blue}{\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right)} + -1\right)\right)\right) \]
9. Step-by-step derivation

   1. distribute-lft-in 99.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\color{blue}{4 \cdot \left({v}^{2} + {v}^{4}\right)} + -1\right)\right)\right) \]

   2. unpow2 99.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \left(\color{blue}{v \cdot v} + {v}^{4}\right) + -1\right)\right)\right) \]

   3. fma-undefine 99.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \color{blue}{\mathsf{fma}\left(v, v, {v}^{4}\right)} + -1\right)\right)\right) \]

10. Simplified 99.0%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\color{blue}{4 \cdot \mathsf{fma}\left(v, v, {v}^{4}\right)} + -1\right)\right)\right) \]

11. Final simplification 99.0%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(4 \cdot \mathsf{fma}\left(v, v, {v}^{4}\right) + -1\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 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 99.0%

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

3. Final simplification 99.0%

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

Alternative 5: 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 99.0%

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

3. Taylor expanded in v around 0 98.4%

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

4. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))