Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 1.0s

Alternatives: 6

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(2.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(0.5 * N[Log[1 + N[(t$95$0 * x + N[(N[(t$95$0 * x), $MachinePrecision] * 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

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

   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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 100.0

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{1 - x \cdot x}, x, \left(\frac{2}{1 - x \cdot x} \cdot x\right) \cdot x\right)}\right) \]
8. 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, 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. 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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 100.0

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 4: 99.7% accurate, 4.5× speedup

Math

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

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. 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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 100.0

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 5: 99.6% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 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. 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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 100.0

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 6: 99.1% 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. 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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 100.0

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.1%

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

Reproduce

herbie shell --seed 2025107 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))