Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 9

Speedup: 1.0×

Specification

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]

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

3. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-add N/A

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

   5. frac-add N/A

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

   6. lower-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

3. Applied rewrites 100.0%

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

Alternative 3: 99.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 1, x \cdot \left(\left(0.3333333333333333 \cdot x + \left(\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  1.0
  (*
   x
   (*
    (+
     (* 0.3333333333333333 x)
     (* (* (* (fma (* x x) 0.14285714285714285 0.2) x) x) x))
    x))))

C

double code(double x) {
	return fma(x, 1.0, (x * (((0.3333333333333333 * x) + (((fma((x * x), 0.14285714285714285, 0.2) * x) * x) * x)) * x)));
}

Julia

function code(x)
	return fma(x, 1.0, Float64(x * Float64(Float64(Float64(0.3333333333333333 * x) + Float64(Float64(Float64(fma(Float64(x * x), 0.14285714285714285, 0.2) * x) * x) * x)) * x)))
end

Wolfram

code[x_] := N[(x * 1.0 + N[(x * N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] + N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.14285714285714285 + 0.2), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x} \]

5. Taylor expanded in x around 0

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

   1. log-pow-rev N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

9. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right)\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. lift-*.f64 N/A

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

    6. pow2 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f64 N/A

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

    9. pow2 N/A

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

    10. *-commutative N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. distribute-rgt-in N/A

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

    14. lower-+.f64 N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 N/A

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

11. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(x, 1, x \cdot \left(\left(0.3333333333333333 \cdot x + \left(\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \]
12. Add Preprocessing

Alternative 4: 99.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (*
   (* (fma (fma (* x x) 0.14285714285714285 0.2) (* x x) 0.3333333333333333) x)
   x)
  x
  x))

C

double code(double x) {
	return fma(((fma(fma((x * x), 0.14285714285714285, 0.2), (x * x), 0.3333333333333333) * x) * x), x, x);
}

Julia

function code(x)
	return fma(Float64(Float64(fma(fma(Float64(x * x), 0.14285714285714285, 0.2), Float64(x * x), 0.3333333333333333) * x) * x), x, x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.14285714285714285 + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x} \]

5. Taylor expanded in x around 0

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

   1. log-pow-rev N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

9. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right)\right) \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

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

    2. *-rgt-identity N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-fma.f64 N/A

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

    7. lift-fma.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. +-commutative N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f64 N/A

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

11. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, x\right) \]
12. Add Preprocessing

Alternative 5: 99.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma (fma 0.14285714285714285 (* x x) 0.2) (* x x) 0.3333333333333333)
   (* x x)
   1.0)
  x))

C

double code(double x) {
	return fma(fma(fma(0.14285714285714285, (x * x), 0.2), (x * x), 0.3333333333333333), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(fma(0.14285714285714285, Float64(x * x), 0.2), Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.14285714285714285 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x} \]

5. Taylor expanded in x around 0

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

   1. log-pow-rev N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
8. Add Preprocessing

Alternative 6: 99.7% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x} \]

5. Taylor expanded in x around 0

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

   1. log-pow-rev N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 99.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

9. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x\right)\right) \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

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

    2. *-rgt-identity N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-fma.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. +-commutative N/A

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

11. Applied rewrites 99.7%

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

Alternative 7: 99.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma (fma 0.2 (* x x) 0.3333333333333333) (* x x) 1.0) x))

C

double code(double x) {
	return fma(fma(0.2, (x * x), 0.3333333333333333), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(0.2, Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x} \]

5. Taylor expanded in x around 0

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

   1. log-pow-rev N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 99.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
8. Add Preprocessing

Alternative 8: 99.5% 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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. cube-mult N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 9: 99.0% accurate, 125.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025095 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))