Hyperbolic arc-(co)tangent

Percentage Accurate: 94.0% → 97.9%

Time: 33.5s

Alternatives: 4

Speedup: 18.5×

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: 94.0% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(x\_m\right) - \mathsf{log1p}\left(-x\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 5.1e-29) 0.0 (* 0.5 (- (log1p x_m) (log1p (- x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.1e-29) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (log1p(x_m) - log1p(-x_m));
	}
	return x_s * tmp;
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.1e-29) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (Math.log1p(x_m) - Math.log1p(-x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5.1e-29:
		tmp = 0.0
	else:
		tmp = 0.5 * (math.log1p(x_m) - math.log1p(-x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5.1e-29)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * Float64(log1p(x_m) - log1p(Float64(-x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5.1e-29], 0.0, N[(0.5 * N[(N[Log[1 + x$95$m], $MachinePrecision] - N[Log[1 + (-x$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.09999999999999986e-29

   1. Initial program 95.9%

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

      1. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. *-commutative 95.9%

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

      3. log-prod 95.9%

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

      4. sub-neg 95.9%

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

      5. add-sqr-sqrt 44.9%

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

      6. sqrt-unprod 95.9%

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

      7. sqr-neg 95.9%

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

      8. sqrt-unprod 51.0%

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

      9. add-sqr-sqrt 94.5%

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

      10. diff-log 94.5%

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

      11. log1p-undefine 6.1%

          \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 + x\right)\right) + \log 1\right) \]

      12. log1p-undefine 94.5%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \log 1\right) \]

      13. metadata-eval 94.5%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(x\right)\right) + \color{blue}{0}\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(x\right)\right) + 0\right)} \]
   7. Step-by-step derivation

      1. +-rgt-identity 94.5%

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

      2. +-inverses 94.5%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

   8. Simplified 94.5%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   if 5.09999999999999986e-29 < x

   1. Initial program 34.5%

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

   3. Simplified 84.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.1e-29) 0.0 (+ x_m (* (pow x_m 3.0) 0.3333333333333333)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.1e-29) {
		tmp = 0.0;
	} else {
		tmp = x_m + (pow(x_m, 3.0) * 0.3333333333333333);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5.1d-29) then
        tmp = 0.0d0
    else
        tmp = x_m + ((x_m ** 3.0d0) * 0.3333333333333333d0)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.1e-29) {
		tmp = 0.0;
	} else {
		tmp = x_m + (Math.pow(x_m, 3.0) * 0.3333333333333333);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5.1e-29:
		tmp = 0.0
	else:
		tmp = x_m + (math.pow(x_m, 3.0) * 0.3333333333333333)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5.1e-29)
		tmp = 0.0;
	else
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * 0.3333333333333333));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5.1e-29)
		tmp = 0.0;
	else
		tmp = x_m + ((x_m ^ 3.0) * 0.3333333333333333);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5.1e-29], 0.0, N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.09999999999999986e-29

   1. Initial program 95.9%

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

      1. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. *-commutative 95.9%

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

      3. log-prod 95.9%

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

      4. sub-neg 95.9%

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

      5. add-sqr-sqrt 44.9%

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

      6. sqrt-unprod 95.9%

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

      7. sqr-neg 95.9%

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

      8. sqrt-unprod 51.0%

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

      9. add-sqr-sqrt 94.5%

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

      10. diff-log 94.5%

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

      11. log1p-undefine 6.1%

          \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 + x\right)\right) + \log 1\right) \]

      12. log1p-undefine 94.5%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \log 1\right) \]

      13. metadata-eval 94.5%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(x\right)\right) + \color{blue}{0}\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(x\right)\right) + 0\right)} \]
   7. Step-by-step derivation

      1. +-rgt-identity 94.5%

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

      2. +-inverses 94.5%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

   8. Simplified 94.5%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   if 5.09999999999999986e-29 < x

   1. Initial program 34.5%

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

      1. metadata-eval 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in x around 0 80.8%

      \[\leadsto \color{blue}{x + 0.3333333333333333 \cdot {x}^{3}} \]
   6. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto x + \color{blue}{{x}^{3} \cdot 0.3333333333333333} \]

   7. Simplified 80.8%

      \[\leadsto \color{blue}{x + {x}^{3} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{3} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (if (<= x_m 5.1e-29) 0.0 x_m)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.1e-29) {
		tmp = 0.0;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5.1d-29) then
        tmp = 0.0d0
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.1e-29) {
		tmp = 0.0;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5.1e-29:
		tmp = 0.0
	else:
		tmp = x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5.1e-29)
		tmp = 0.0;
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5.1e-29)
		tmp = 0.0;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5.1e-29], 0.0, x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.09999999999999986e-29

   1. Initial program 95.9%

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

      1. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. *-commutative 95.9%

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

      3. log-prod 95.9%

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

      4. sub-neg 95.9%

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

      5. add-sqr-sqrt 44.9%

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

      6. sqrt-unprod 95.9%

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

      7. sqr-neg 95.9%

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

      8. sqrt-unprod 51.0%

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

      9. add-sqr-sqrt 94.5%

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

      10. diff-log 94.5%

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

      11. log1p-undefine 6.1%

          \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 + x\right)\right) + \log 1\right) \]

      12. log1p-undefine 94.5%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \log 1\right) \]

      13. metadata-eval 94.5%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(x\right)\right) + \color{blue}{0}\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(x\right)\right) + 0\right)} \]
   7. Step-by-step derivation

      1. +-rgt-identity 94.5%

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

      2. +-inverses 94.5%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

   8. Simplified 94.5%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   if 5.09999999999999986e-29 < x

   1. Initial program 34.5%

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

      1. metadata-eval 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in x around 0 78.2%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 11.4% accurate, 111.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. metadata-eval 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in x around 0 11.2%

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

6. Final simplification 11.2%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024047 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))