Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

Alternatives: 8

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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(N[(x * -2.0), $MachinePrecision] / N[(-1.0 + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x + N[(N[(x * x), $MachinePrecision] + 1.0), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. flip3-+ 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{-2 \cdot x}{\color{blue}{\frac{{x}^{3} + {-1}^{3}}{x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)}}}\right) \]

   6. associate-/r/ 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-2 \cdot x}{{x}^{3} + {-1}^{3}} \cdot \left(x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)\right)}\right) \]

   7. *-commutative 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{-\color{blue}{x \cdot 2}}{{x}^{3} + {-1}^{3}} \cdot \left(x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)\right)\right) \]

   8. distribute-rgt-neg-in 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot \left(-2\right)}}{{x}^{3} + {-1}^{3}} \cdot \left(x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)\right)\right) \]

   9. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot \color{blue}{-2}}{{x}^{3} + {-1}^{3}} \cdot \left(x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot -2}{{x}^{3} + \color{blue}{-1}} \cdot \left(x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)\right)\right) \]

   11. +-commutative 100.0%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot -2}{\color{blue}{-1 + {x}^{3}}} \cdot \left(x \cdot x + \left(-1 \cdot -1 - x \cdot -1\right)\right)\right) \]

   12. cube-mult 100.0%

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

   13. metadata-eval 100.0%

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

   14. *-commutative 100.0%

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

   15. neg-mul-1 100.0%

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

   16. neg-sub0 100.0%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot -2}{-1 + x \cdot \left(x \cdot x\right)} \cdot \left(x \cdot x + \left(1 - \color{blue}{\left(0 - x\right)}\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{x \cdot -2}{-1 + x \cdot \left(x \cdot x\right)} \cdot \left(x \cdot x + \left(1 - \left(0 - x\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-+r- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot -2}{-1 + x \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(\left(x \cdot x + 1\right) - \left(0 - x\right)\right)}\right) \]

   2. sub-neg 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot -2}{-1 + x \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(\left(x \cdot x + 1\right) + \left(-\left(0 - x\right)\right)\right)}\right) \]

   3. sub0-neg 100.0%

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

   4. remove-double-neg 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(x * 2.0), $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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(x * N[(-2.0 / N[(x + -1.0), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 4: 99.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x * (2.0d0 + (x * (x * (0.6666666666666666d0 + ((x * x) * (0.4d0 + ((x * x) * 0.2857142857142857d0))))))))
end function

Java

public static double code(double x) {
	return 0.5 * (x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.4 + ((x * x) * 0.2857142857142857))))))));
}

Python

def code(x):
	return 0.5 * (x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.4 + ((x * x) * 0.2857142857142857))))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x * (2.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.4 + ((x * x) * 0.2857142857142857))))))));
end

Wolfram

code[x_] := N[(0.5 * N[(x * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.4 + N[(N[(x * x), $MachinePrecision] * 0.2857142857142857), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right) \]

   2. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{x \cdot \left(x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.4 + 0.2857142857142857 \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. *-commutative 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \color{blue}{{x}^{2} \cdot 0.2857142857142857}\right)\right)\right)\right)\right) \]

   5. unpow2 99.5%

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

7. Simplified 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.4 + \left(x \cdot x\right) \cdot 0.2857142857142857\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 99.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x * (2.0d0 + (x * (x * (0.6666666666666666d0 + (x * (x * 0.4d0)))))))
end function

Java

public static double code(double x) {
	return 0.5 * (x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.4)))))));
}

Python

def code(x):
	return 0.5 * (x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.4)))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x * (2.0 + (x * (x * (0.6666666666666666 + (x * (x * 0.4)))))));
end

Wolfram

code[x_] := N[(0.5 * N[(x * N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(x * N[(x * 0.4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 99.5%

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

   1. unpow2 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.6666666666666666 + 0.4 \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{x \cdot \left(x \cdot \left(0.6666666666666666 + 0.4 \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. *-commutative 99.5%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + \color{blue}{{x}^{2} \cdot 0.4}\right)\right)\right)\right) \]

   4. unpow2 99.5%

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

   5. associate-*l* 99.5%

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

7. Simplified 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot 0.4\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 6: 99.6% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * 2.0d0) + (x * ((x * x) * 0.6666666666666666d0)))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * 2.0) + (x * ((x * x) * 0.6666666666666666)));
}

Python

def code(x):
	return 0.5 * ((x * 2.0) + (x * ((x * x) * 0.6666666666666666)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * 2.0) + (x * ((x * x) * 0.6666666666666666)));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * 2.0), $MachinePrecision] + N[(x * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 99.2%

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

   1. *-commutative 99.2%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{{x}^{2} \cdot 0.6666666666666666}\right)\right) \]

   2. unpow2 99.2%

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

   3. associate-*l* 99.2%

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

7. Simplified 99.2%

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

   1. distribute-lft-in 99.2%

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

   2. associate-*r* 99.2%

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

9. Applied egg-rr 99.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)} \]
10. Add Preprocessing

Alternative 7: 99.6% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x * (2.0d0 + (x * (x * 0.6666666666666666d0))))
end function

Java

public static double code(double x) {
	return 0.5 * (x * (2.0 + (x * (x * 0.6666666666666666))));
}

Python

def code(x):
	return 0.5 * (x * (2.0 + (x * (x * 0.6666666666666666))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x * (2.0 + (x * (x * 0.6666666666666666))));
end

Wolfram

code[x_] := N[(0.5 * N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 99.2%

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

   1. *-commutative 99.2%

      \[\leadsto 0.5 \cdot \left(x \cdot \left(2 + \color{blue}{{x}^{2} \cdot 0.6666666666666666}\right)\right) \]

   2. unpow2 99.2%

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

   3. associate-*l* 99.2%

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

7. Simplified 99.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)} \]
8. Add Preprocessing

Alternative 8: 99.1% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(x * 2.0), $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. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{\left(0 - x\right)} + 1}\right) \]

   6. associate-+l- 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.8%

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

6. Final simplification 98.8%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))