Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \tanh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (atanh x))

C

double code(double x) {
	return atanh(x);
}

Python

def code(x):
	return math.atanh(x)

Julia

function code(x)
	return atanh(x)
end

MATLAB

function tmp = code(x)
	tmp = atanh(x);
end

Wolfram

code[x_] := N[ArcTanh[x], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

double code(double x) {
	return 0.5 * log1p(((2.0 * x) / (1.0 - x)));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p(((2.0 * x) / (1.0 - x)));
}

Python

def code(x):
	return 0.5 * math.log1p(((2.0 * x) / (1.0 - x)))

Julia

function code(x)
	return Float64(0.5 * log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left(\mathsf{fma}\left(x, x, x\right) \cdot 2\right) \cdot {\left(\mathsf{fma}\left(-x, x, 1\right)\right)}^{-1}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (log1p (* (* (fma x x x) 2.0) (pow (fma (- x) x 1.0) -1.0))) 0.5))

C

double code(double x) {
	return log1p(((fma(x, x, x) * 2.0) * pow(fma(-x, x, 1.0), -1.0))) * 0.5;
}

Julia

function code(x)
	return Float64(log1p(Float64(Float64(fma(x, x, x) * 2.0) * (fma(Float64(-x), x, 1.0) ^ -1.0))) * 0.5)
end

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(N[(x * x + x), $MachinePrecision] * 2.0), $MachinePrecision] * N[Power[N[((-x) * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 - x}}\right) \]

   2. clear-num N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{1 - x}{2 \cdot x}}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{1 - x}}{2 \cdot x}}\right) \]

   4. flip-- N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}}}{2 \cdot x}}\right) \]

   5. associate-/l/ N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{\left(2 \cdot x\right) \cdot \left(1 + x\right)}}}\right) \]

   6. associate-/r/ N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)}\right) \]

   8. inv-pow N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{{\left(1 \cdot 1 - x \cdot x\right)}^{-1}} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{{\left(1 \cdot 1 - x \cdot x\right)}^{-1}} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left({\left(\color{blue}{1} - x \cdot x\right)}^{-1} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left({\color{blue}{\left(1 + \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}}^{-1} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left({\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + 1\right)}}^{-1} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left({\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot x} + 1\right)}^{-1} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(x\right), x, 1\right)\right)}}^{-1} \cdot \left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)\right) \]

   15. lower-neg.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. +-commutative N/A

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

   19. distribute-rgt-out N/A

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

   20. lower-*.f64 N/A

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

   21. *-lft-identity N/A

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

   22. lower-fma.f64 100.0

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left({\left(\mathsf{fma}\left(-x, x, 1\right)\right)}^{-1} \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(x, x, x\right)}\right)\right) \]

4. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(-x, x, 1\right)\right)}^{-1} \cdot \left(2 \cdot \mathsf{fma}\left(x, x, x\right)\right)}\right) \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(\left(\mathsf{fma}\left(x, x, x\right) \cdot 2\right) \cdot {\left(\mathsf{fma}\left(-x, x, 1\right)\right)}^{-1}\right) \cdot 0.5 \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (log1p (/ (* 2.0 x) (- 1.0 x))) 0.5))

C

double code(double x) {
	return log1p(((2.0 * x) / (1.0 - x))) * 0.5;
}

Java

public static double code(double x) {
	return Math.log1p(((2.0 * x) / (1.0 - x))) * 0.5;
}

Python

def code(x):
	return math.log1p(((2.0 * x) / (1.0 - x))) * 0.5

Julia

function code(x)
	return Float64(log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))) * 0.5)
end

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot 0.5 \]
4. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{-2}{x - 1} \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (log1p (* (/ -2.0 (- x 1.0)) x)) 0.5))

C

double code(double x) {
	return log1p(((-2.0 / (x - 1.0)) * x)) * 0.5;
}

Java

public static double code(double x) {
	return Math.log1p(((-2.0 / (x - 1.0)) * x)) * 0.5;
}

Python

def code(x):
	return math.log1p(((-2.0 / (x - 1.0)) * x)) * 0.5

Julia

function code(x)
	return Float64(log1p(Float64(Float64(-2.0 / Float64(x - 1.0)) * x)) * 0.5)
end

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(-2.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 - x}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{1 - x}\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{1 - x}\right) \]

   4. associate-/l* N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{x \cdot \frac{2}{1 - x}}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2}{1 - x} \cdot x}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2}{1 - x} \cdot x}\right) \]

   7. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}} \cdot x\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}} \cdot x\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{-2}}{\mathsf{neg}\left(\left(1 - x\right)\right)} \cdot x\right) \]

   10. neg-sub0 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{0 - \left(1 - x\right)}} \cdot x\right) \]

   11. lift--.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{0 - \color{blue}{\left(1 - x\right)}} \cdot x\right) \]

   12. sub-neg N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}} \cdot x\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)}} \cdot x\right) \]

   14. associate--r+ N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - 1}} \cdot x\right) \]

   15. neg-sub0 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} - 1} \cdot x\right) \]

   16. remove-double-neg N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{x} - 1} \cdot x\right) \]

   17. lower--.f64 100.0

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{x - 1}} \cdot x\right) \]

4. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\frac{-2}{x - 1} \cdot x\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(\frac{-2}{x - 1} \cdot x\right) \cdot 0.5 \]
6. Add Preprocessing

Alternative 4: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (*
    (* (fma (fma 0.2857142857142857 (* x x) 0.4) (* x x) 0.6666666666666666) x)
    x)
   x
   (* 2.0 x))
  0.5))

C

double code(double x) {
	return fma(((fma(fma(0.2857142857142857, (x * x), 0.4), (x * x), 0.6666666666666666) * x) * x), x, (2.0 * x)) * 0.5;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(fma(0.2857142857142857, Float64(x * x), 0.4), Float64(x * x), 0.6666666666666666) * x) * x), x, Float64(2.0 * x)) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.2857142857142857 * N[(x * x), $MachinePrecision] + 0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

4. Applied rewrites 7.9%

   \[\leadsto 0.5 \cdot \color{blue}{\log \left(\mathsf{fma}\left(\frac{-2}{x - 1}, x, 1\right)\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{2}{3}, {x}^{2}, 2\right) \cdot x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{7}, {x}^{2}, \frac{2}{5}\right)}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, \color{blue}{x \cdot x}, \frac{2}{5}\right), {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, \color{blue}{x \cdot x}, \frac{2}{5}\right), {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   15. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   16. lower-*.f64 99.6

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

7. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.7%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right) \]

10. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 5: 99.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma
    (fma (fma 0.2857142857142857 (* x x) 0.4) (* x x) 0.6666666666666666)
    (* x x)
    2.0)
   x)
  0.5))

C

double code(double x) {
	return (fma(fma(fma(0.2857142857142857, (x * x), 0.4), (x * x), 0.6666666666666666), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(fma(0.2857142857142857, Float64(x * x), 0.4), Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.2857142857142857 * N[(x * x), $MachinePrecision] + 0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{2}{3}, {x}^{2}, 2\right) \cdot x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5} + \frac{2}{7} \cdot {x}^{2}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{2}{7} \cdot {x}^{2} + \frac{2}{5}}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{7}, {x}^{2}, \frac{2}{5}\right)}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, \color{blue}{x \cdot x}, \frac{2}{5}\right), {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, \color{blue}{x \cdot x}, \frac{2}{5}\right), {x}^{2}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   15. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{7}, x \cdot x, \frac{2}{5}\right), x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   16. lower-*.f64 99.6

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]

6. Final simplification 99.6%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 6: 99.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(0.4 \cdot \left(x \cdot x\right) + 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* (* (+ (* 0.4 (* x x)) 0.6666666666666666) x) x) x (* 2.0 x)) 0.5))

C

double code(double x) {
	return fma(((((0.4 * (x * x)) + 0.6666666666666666) * x) * x), x, (2.0 * x)) * 0.5;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(Float64(Float64(0.4 * Float64(x * x)) + 0.6666666666666666) * x) * x), x, Float64(2.0 * x)) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{5} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   11. lower-*.f64 99.6

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.4, 0.6666666666666666\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\left(0.4 \cdot \left(x \cdot x\right) + 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \]

10. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left(\left(\left(0.4 \cdot \left(x \cdot x\right) + 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 7: 99.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.4, 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (* (* (fma (* x x) 0.4 0.6666666666666666) x) x) x (* 2.0 x)) 0.5))

C

double code(double x) {
	return fma(((fma((x * x), 0.4, 0.6666666666666666) * x) * x), x, (2.0 * x)) * 0.5;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(Float64(x * x), 0.4, 0.6666666666666666) * x) * x), x, Float64(2.0 * x)) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.4 + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{5} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   11. lower-*.f64 99.6

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.4, 0.6666666666666666\right) \cdot x\right) \cdot x, \color{blue}{x}, 2 \cdot x\right) \]

8. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.4, 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \]
9. Add Preprocessing

Alternative 8: 99.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma (fma 0.4 (* x x) 0.6666666666666666) (* x x) 2.0) x) 0.5))

C

double code(double x) {
	return (fma(fma(0.4, (x * x), 0.6666666666666666), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(0.4, Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right)\right) \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) + 2\right)} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) \cdot x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{2}{5} \cdot {x}^{2}, {x}^{2}, 2\right)} \cdot x\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{5} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, 2\right) \cdot x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{5}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right) \cdot x\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right) \cdot x\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{5}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

   11. lower-*.f64 99.6

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]

6. Final simplification 99.6%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 9: 99.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* (* x x) x) 0.3333333333333333 x))

C

double code(double x) {
	return fma(((x * x) * x), 0.3333333333333333, x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * x), 0.3333333333333333, x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{3}\right)} + x \cdot 1 \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{1}{3}} + x \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{1}{3} + x \cdot 1 \]

   6. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{3} + x \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \frac{1}{3} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{3}, x\right)} \]

   9. lower-pow.f64 99.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, 0.3333333333333333, x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.3333333333333333, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) \]
8. Add Preprocessing

Alternative 10: 99.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 2.0 x) 0.5))

C

double code(double x) {
	return (2.0 * x) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 * x) * 0.5d0
end function

Java

public static double code(double x) {
	return (2.0 * x) * 0.5;
}

Python

def code(x):
	return (2.0 * x) * 0.5

Julia

function code(x)
	return Float64(Float64(2.0 * x) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * x) * 0.5;
end

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot x\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 99.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x\right)} \]

5. Applied rewrites 99.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x\right)} \]

6. Final simplification 99.0%

   \[\leadsto \left(2 \cdot x\right) \cdot 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024264 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))