Hyperbolic arc-(co)tangent

Percentage Accurate: 8.4% → 99.5%

Time: 12.5s

Alternatives: 3

Speedup: 22.3×

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: 99.5% 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 6.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 100.0

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

5. Simplified 100.0%

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

Alternative 2: 17.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * x
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * x;
}

Python

def code(x):
	return 0.3333333333333333 * x

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * x;
end

Wolfram

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

Derivation

1. Initial program 6.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 100.0

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

5. Simplified 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot {x}^{3}} \]
7. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 6.2

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

8. Simplified 6.2%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \left(x \cdot x\right)\right)} \]
9. Step-by-step derivation

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqr-pow N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. rem-exp-log N/A

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

   7. sqr-pow N/A

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

   8. metadata-eval N/A

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

   9. unpow1 N/A

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

   10. rem-exp-log N/A

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

   11. prod-exp N/A

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

   12. flip-+ N/A

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

   13. +-inverses N/A

       \[\leadsto x \cdot \left(\frac{1}{3} \cdot e^{\frac{\color{blue}{0}}{\log x - \log x}}\right) \]

   14. +-inverses N/A

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

   15. +-inverses N/A

       \[\leadsto x \cdot \left(\frac{1}{3} \cdot e^{\frac{\left(\log x \cdot \frac{\frac{2}{2}}{2}\right) \cdot \left(\log x \cdot \frac{\frac{2}{2}}{2}\right) - \left(\log x \cdot \frac{\frac{2}{2}}{2}\right) \cdot \left(\log x \cdot \frac{\frac{2}{2}}{2}\right)}{\color{blue}{0}}}\right) \]

   16. +-inverses N/A

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

   17. flip-+ N/A

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

   18. distribute-lft-out N/A

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

10. Applied egg-rr 17.7%

    \[\leadsto x \cdot \color{blue}{0.3333333333333333} \]

11. Final simplification 17.7%

    \[\leadsto 0.3333333333333333 \cdot x \]
12. Add Preprocessing

Alternative 3: 3.8% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.3333333333333333)

C

double code(double x) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return 0.3333333333333333;
}

Python

def code(x):
	return 0.3333333333333333

Julia

function code(x)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_] := 0.3333333333333333

Derivation

1. Initial program 6.6%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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 100.0

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

5. Simplified 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot {x}^{3}} \]
7. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 6.2

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

8. Simplified 6.2%

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

10. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))