Hyperbolic arc-(co)tangent

Percentage Accurate: 8.5% → 100.0%

Time: 6.5s

Alternatives: 6

Speedup: 12.2×

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left({x}^{3}\right) - \left(\mathsf{log1p}\left(x \cdot x - x\right) + \mathsf{log1p}\left(-x\right)\right)\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- (log1p (pow x 3.0)) (+ (log1p (- (* x x) x)) (log1p (- x)))) 0.5))

C

double code(double x) {
	return (log1p(pow(x, 3.0)) - (log1p(((x * x) - x)) + log1p(-x))) * 0.5;
}

Java

public static double code(double x) {
	return (Math.log1p(Math.pow(x, 3.0)) - (Math.log1p(((x * x) - x)) + Math.log1p(-x))) * 0.5;
}

Python

def code(x):
	return (math.log1p(math.pow(x, 3.0)) - (math.log1p(((x * x) - x)) + math.log1p(-x))) * 0.5

Julia

function code(x)
	return Float64(Float64(log1p((x ^ 3.0)) - Float64(log1p(Float64(Float64(x * x) - x)) + log1p(Float64(-x)))) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[Log[1 + N[Power[x, 3.0], $MachinePrecision]], $MachinePrecision] - N[(N[Log[1 + N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 7.9

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

4. Applied rewrites 7.9%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-+.f64 N/A

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

   5. flip3-+ N/A

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

   6. associate-*l/ N/A

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

   7. log-div N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval N/A

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

6. Applied rewrites 20.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\frac{{x}^{3} + 1}{1 - x}\right) - \mathsf{log1p}\left(x \cdot x - x\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. log-div N/A

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

   5. associate--l- N/A

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

   6. lower--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-log1p.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \left(\mathsf{log1p}\left({x}^{3}\right) - \left(\mathsf{log1p}\left(x \cdot x - x\right) + \mathsf{log1p}\left(-x\right)\right)\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2857142857142857 + 0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.5

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, \color{blue}{x}, x \cdot 2\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.5

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2857142857142857, 0.4\right), x \cdot x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, 2 \cdot x\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2857142857142857 + 0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.5

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.5

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(x \cdot x, \frac{2}{7}, \frac{2}{5}\right)\right)\right), x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 99.5%

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

Alternative 4: 99.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(0.4 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.4

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.4, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

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

   5. metadata-eval 99.4

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

7. Applied rewrites 99.4%

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

Alternative 5: 99.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.2

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{fma}\left(0.6666666666666666, \color{blue}{x \cdot x}, 2\right) \cdot x\right) \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.2

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-/.f64 N/A

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

   5. metadata-eval 99.2

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

7. Applied rewrites 99.2%

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

Alternative 6: 99.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.9

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

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(x \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{Rewrite=>}\left(lower-*.f64, \left(x \cdot 2\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))