Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 7

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(x + 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. count-2-rev N/A

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

   3. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

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 99.2

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 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 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 99.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (fma 0.2 (* x x) 0.3333333333333333) (* x x)) x x))

C

double code(double x) {
	return fma((fma(0.2, (x * x), 0.3333333333333333) * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(fma(0.2, Float64(x * x), 0.3333333333333333) * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + 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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. cube-mult N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.1

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.2, \color{blue}{x \cdot x}, 0.3333333333333333\right), x\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
8. Add Preprocessing

Alternative 5: 99.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(0.6666666666666666 * 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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Final simplification 98.8%

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

Alternative 6: 99.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. cube-mult N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.1

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.2, \color{blue}{x \cdot x}, 0.3333333333333333\right), x\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot \left(x \cdot x\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 98.8%

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

Alternative 7: 99.0% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + 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 98.2

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

   \[\leadsto \left(x + x\right) \cdot 0.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))