Hyperbolic arc-(co)tangent

Percentage Accurate: 8.4% → 99.8%

Time: 11.8s

Alternatives: 5

Speedup: 134.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / 2.0d0) * log(((1.0d0 + x) / (1.0d0 - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

Wolfram

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

Initial Program: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / 2.0d0) * log(((1.0d0 + x) / (1.0d0 - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

Wolfram

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

Alternative 1: 99.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.04656084656084656, -0.08888888888888889\right), -0.3333333333333333\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  x
  (fma
   (* x x)
   (fma
    x
    (* x (fma x (* x -0.04656084656084656) -0.08888888888888889))
    -0.3333333333333333)
   1.0)))

C

double code(double x) {
	return x / fma((x * x), fma(x, (x * fma(x, (x * -0.04656084656084656), -0.08888888888888889)), -0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x / fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.04656084656084656), -0.08888888888888889)), -0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.04656084656084656), $MachinePrecision] + -0.08888888888888889), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}\right)}{x}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}\right) + 1}}{x}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}, 1\right)}}{x}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}, 1\right)}{x}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}, 1\right)}{x}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)}{x}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) + \color{blue}{\frac{-1}{3}}, 1\right)}{x}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}, \frac{-1}{3}\right)}, 1\right)}{x}} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}, \frac{-1}{3}\right), 1\right)}{x}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}, \frac{-1}{3}\right), 1\right)}{x}} \]

   11. sub-neg N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-44}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{4}{45}\right)\right)}, \frac{-1}{3}\right), 1\right)}{x}} \]

   12. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-44}{945}} + \left(\mathsf{neg}\left(\frac{4}{45}\right)\right), \frac{-1}{3}\right), 1\right)}{x}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-44}{945} + \color{blue}{\frac{-4}{45}}, \frac{-1}{3}\right), 1\right)}{x}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-44}{945}, \frac{-4}{45}\right)}, \frac{-1}{3}\right), 1\right)}{x}} \]

   15. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-44}{945}, \frac{-4}{45}\right), \frac{-1}{3}\right), 1\right)}{x}} \]

   16. lower-*.f64 99.2

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.04656084656084656, -0.08888888888888889\right), -0.3333333333333333\right), 1\right)}{x}} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.04656084656084656, -0.08888888888888889\right), -0.3333333333333333\right), 1\right)}{x}}} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-44}{945} + \frac{-4}{45}\right) + \frac{-1}{3}\right) + 1}{x}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{-44}{945} + \frac{-4}{45}\right) + \frac{-1}{3}\right) + 1}{x}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-44}{945} + \frac{-4}{45}\right) + \frac{-1}{3}\right) + 1}{x}} \]

    4. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-44}{945}, \frac{-4}{45}\right)} + \frac{-1}{3}\right) + 1}{x}} \]

    5. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-44}{945}, \frac{-4}{45}\right), \frac{-1}{3}\right)} + 1}{x}} \]

    6. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-44}{945}, \frac{-4}{45}\right), \frac{-1}{3}\right), 1\right)}}{x}} \]

    7. clear-num N/A

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-44}{945}, \frac{-4}{45}\right), \frac{-1}{3}\right), 1\right)}} \]

    8. lower-/.f64 99.5

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.04656084656084656, -0.08888888888888889\right), -0.3333333333333333\right), 1\right)}} \]

12. Applied rewrites 99.5%

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.04656084656084656, -0.08888888888888889\right), -0.3333333333333333\right), 1\right)}} \]
13. Add Preprocessing

Alternative 2: 99.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. cube-mult N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. lower-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 5: 99.1% accurate, 134.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    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 8.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval 8.6

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

4. Applied rewrites 8.6%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.8

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

7. Applied rewrites 98.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

      \[\leadsto \color{blue}{1} \cdot x \]

   4. *-lft-identity 98.8

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

9. Applied rewrites 98.8%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))