Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 9

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{fma}\left(-x, x, 1\right)} \cdot 2\\ \mathsf{log1p}\left(\mathsf{fma}\left(t\_0, x, t\_0\right)\right) \cdot 0.5 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (/ x (fma (- x) x 1.0)) 2.0)))
   (* (log1p (fma t_0 x t_0)) 0.5)))

C

double code(double x) {
	double t_0 = (x / fma(-x, x, 1.0)) * 2.0;
	return log1p(fma(t_0, x, t_0)) * 0.5;
}

Julia

function code(x)
	t_0 = Float64(Float64(x / fma(Float64(-x), x, 1.0)) * 2.0)
	return Float64(log1p(fma(t_0, x, t_0)) * 0.5)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[((-x) * x + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(N[Log[1 + N[(t$95$0 * x + t$95$0), $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

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

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

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(2.0 * x + 2.0), $MachinePrecision] * N[(x / N[((-x) * x + 1.0), $MachinePrecision]), $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. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

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

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2 \cdot \frac{x}{\mathsf{fma}\left(-x, x, 1\right)}, x, \left(2 \cdot \frac{x}{\mathsf{fma}\left(-x, x, 1\right)}\right) \cdot 1\right)}\right) \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-rgt-identity N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{x}{\mathsf{fma}\left(-x, x, 1\right)} \cdot \left(2 \cdot \left(1 + x\right)\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(x * 2.0), $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%

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

Alternative 4: 99.7% 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 5: 99.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(x * 2.0), $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 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 6: 99.6% 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 99.4

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

5. Applied rewrites 99.4%

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

6. Final simplification 99.4%

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

Alternative 7: 99.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right)} \]
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 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 8: 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 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 9: 99.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * 2.0), $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 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))