Hyperbolic arc-(co)tangent

Percentage Accurate: 8.4% → 99.8%

Time: 13.5s

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, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
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-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. cube-mult N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

5. Simplified 99.5%

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

Alternative 2: 99.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma (* x x) (fma x (* x 0.2) 0.3333333333333333) 1.0)))

C

double code(double x) {
	return x * fma((x * x), fma(x, (x * 0.2), 0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
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-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. cube-mult N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.4

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.4

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

8. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.3333333333333333\right), 1\right)} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.0

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

5. Simplified 99.0%

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

   1. lift-*.f64 N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.0

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

7. Applied egg-rr 99.0%

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

Alternative 4: 99.5% accurate, 7.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (fma (* x x) 0.3333333333333333 1.0)))

C

double code(double x) {
	return x * fma((x * x), 0.3333333333333333, 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), 0.3333333333333333, 1.0))
end

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.0

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

5. Simplified 99.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right)} \]
6. Add Preprocessing

Alternative 5: 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 7.9%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

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

8. Simplified 98.5%

   \[\leadsto x \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 98.5

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

10. Applied egg-rr 98.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))