Hyperbolic arc-(co)tangent

Percentage Accurate: 8.5% → 100.0%

Time: 12.7s

Alternatives: 6

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: 8.5% 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: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (-
   (* 2.0 (log1p x))
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.25 -0.3333333333333333) -0.5)
     -1.0)))
  0.5))

C

double code(double x) {
	return ((2.0 * log1p(x)) - ((x * x) * fma((x * x), fma((x * x), fma((x * x), -0.25, -0.3333333333333333), -0.5), -1.0))) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * log1p(x)) - Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.25, -0.3333333333333333), -0.5), -1.0))) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(2.0 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}, -1\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}, -1\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}, \frac{-1}{2}\right), -1\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}, \frac{-1}{2}\right), -1\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   17. lower-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)\right) \]

   19. lower-*.f64 99.7

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-log1p.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)} - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2 \cdot \mathsf{log1p}\left(x\right)} - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{4}, \frac{-1}{3}\right)} + \frac{-1}{2}\right) + -1\right)\right) \]

   9. lift-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)} + -1\right)\right) \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)}\right) \]

9. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(2 \cdot \mathsf{log1p}\left(x\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot 0.5} \]
10. Add Preprocessing

Alternative 3: 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 7.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 99.7%

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

Alternative 4: 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 7.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 99.6

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

5. Applied rewrites 99.6%

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

Alternative 5: 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 7.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.0% 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 7.6%

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

   1. metadata-eval 7.6

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

4. Applied rewrites 7.6%

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

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 99.1

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

7. Applied rewrites 99.1%

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

   1. metadata-eval N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity 99.1

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

9. Applied rewrites 99.1%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))