Hyperbolic tangent

Percentage Accurate: 8.9% → 97.8%

Time: 6.7s

Alternatives: 8

Speedup: 409.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t_0}{e^{x} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 8.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t_0}{e^{x} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x_m}\\ t_1 := \frac{e^{x_m} - t_0}{e^{x_m} + t_0}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0.0005:\\ \;\;\;\;x_m + \left(-0.3333333333333333 \cdot {x_m}^{3} + 0.13333333333333333 \cdot {x_m}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))) (t_1 (/ (- (exp x_m) t_0) (+ (exp x_m) t_0))))
   (*
    x_s
    (if (<= t_1 0.0005)
      (+
       x_m
       (+
        (* -0.3333333333333333 (pow x_m 3.0))
        (* 0.13333333333333333 (pow x_m 5.0))))
      t_1))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = exp(-x_m);
	double t_1 = (exp(x_m) - t_0) / (exp(x_m) + t_0);
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = x_m + ((-0.3333333333333333 * pow(x_m, 3.0)) + (0.13333333333333333 * pow(x_m, 5.0)));
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-x_m)
    t_1 = (exp(x_m) - t_0) / (exp(x_m) + t_0)
    if (t_1 <= 0.0005d0) then
        tmp = x_m + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (0.13333333333333333d0 * (x_m ** 5.0d0)))
    else
        tmp = t_1
    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 t_0 = Math.exp(-x_m);
	double t_1 = (Math.exp(x_m) - t_0) / (Math.exp(x_m) + t_0);
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = x_m + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (0.13333333333333333 * Math.pow(x_m, 5.0)));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.exp(-x_m)
	t_1 = (math.exp(x_m) - t_0) / (math.exp(x_m) + t_0)
	tmp = 0
	if t_1 <= 0.0005:
		tmp = x_m + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (0.13333333333333333 * math.pow(x_m, 5.0)))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = exp(Float64(-x_m))
	t_1 = Float64(Float64(exp(x_m) - t_0) / Float64(exp(x_m) + t_0))
	tmp = 0.0
	if (t_1 <= 0.0005)
		tmp = Float64(x_m + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(0.13333333333333333 * (x_m ^ 5.0))));
	else
		tmp = t_1;
	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)
	t_0 = exp(-x_m);
	t_1 = (exp(x_m) - t_0) / (exp(x_m) + t_0);
	tmp = 0.0;
	if (t_1 <= 0.0005)
		tmp = x_m + ((-0.3333333333333333 * (x_m ^ 3.0)) + (0.13333333333333333 * (x_m ^ 5.0)));
	else
		tmp = t_1;
	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_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[x$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 0.0005], N[(x$95$m + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 5.0000000000000001e-4

   1. Initial program 9.3%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

   if 5.0000000000000001e-4 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 65.9%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 0.0005:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \frac{{x_m}^{3} \cdot 0.3333333333333333 + x_m \cdot 2}{2 + \mathsf{fma}\left(x_m, x_m, 0.08333333333333333 \cdot {x_m}^{4}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+ (* (pow x_m 3.0) 0.3333333333333333) (* x_m 2.0))
   (+ 2.0 (fma x_m x_m (* 0.08333333333333333 (pow x_m 4.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((pow(x_m, 3.0) * 0.3333333333333333) + (x_m * 2.0)) / (2.0 + fma(x_m, x_m, (0.08333333333333333 * pow(x_m, 4.0)))));
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64((x_m ^ 3.0) * 0.3333333333333333) + Float64(x_m * 2.0)) / Float64(2.0 + fma(x_m, x_m, Float64(0.08333333333333333 * (x_m ^ 4.0))))))
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 * N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(x$95$m * x$95$m + N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.1%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around 0 97.3%

   \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{\color{blue}{2 + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)}} \]
5. Step-by-step derivation

   1. +-commutative 97.3%

      \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{2 + \color{blue}{\left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

   2. unpow2 97.3%

      \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   3. fma-def 97.3%

      \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{2 + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]

6. Applied egg-rr 97.3%

   \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{2 + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}} \]

7. Final simplification 97.3%

   \[\leadsto \frac{{x}^{3} \cdot 0.3333333333333333 + x \cdot 2}{2 + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]
8. Add Preprocessing

Alternative 3: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \frac{{x_m}^{3} \cdot 0.3333333333333333 + x_m \cdot 2}{\mathsf{fma}\left(x_m, x_m, 2\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/ (+ (* (pow x_m 3.0) 0.3333333333333333) (* x_m 2.0)) (fma x_m x_m 2.0))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((pow(x_m, 3.0) * 0.3333333333333333) + (x_m * 2.0)) / fma(x_m, x_m, 2.0));
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64((x_m ^ 3.0) * 0.3333333333333333) + Float64(x_m * 2.0)) / fma(x_m, x_m, 2.0)))
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 * N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.1%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around 0 97.3%

   \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{\color{blue}{2 + {x}^{2}}} \]
5. Step-by-step derivation

   1. +-commutative 8.5%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{{x}^{2} + 2}} \]

   2. unpow2 8.5%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{x \cdot x} + 2} \]

   3. fma-def 8.5%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

6. Simplified 97.3%

   \[\leadsto \frac{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

7. Final simplification 97.3%

   \[\leadsto \frac{{x}^{3} \cdot 0.3333333333333333 + x \cdot 2}{\mathsf{fma}\left(x, x, 2\right)} \]
8. Add Preprocessing

Alternative 4: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + \left(-0.3333333333333333 \cdot {x_m}^{3} + {x_m}^{5} \cdot 0.125\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+ x_m (+ (* -0.3333333333333333 (pow x_m 3.0)) (* (pow x_m 5.0) 0.125)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m + ((-0.3333333333333333 * pow(x_m, 3.0)) + (pow(x_m, 5.0) * 0.125)));
}

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 + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + ((x_m ** 5.0d0) * 0.125d0)))
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 + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (Math.pow(x_m, 5.0) * 0.125)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (math.pow(x_m, 5.0) * 0.125)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64((x_m ^ 5.0) * 0.125))))
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 + ((-0.3333333333333333 * (x_m ^ 3.0)) + ((x_m ^ 5.0) * 0.125)));
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 * N[(x$95$m + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 5.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.1%

   \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3} + 2 \cdot x}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around 0 97.2%

   \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.125 \cdot {x}^{5}\right)} \]

5. Final simplification 97.2%

   \[\leadsto x + \left(-0.3333333333333333 \cdot {x}^{3} + {x}^{5} \cdot 0.125\right) \]
6. Add Preprocessing

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + \left(-0.3333333333333333 \cdot {x_m}^{3} + 0.13333333333333333 \cdot {x_m}^{5}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   x_m
   (+
    (* -0.3333333333333333 (pow x_m 3.0))
    (* 0.13333333333333333 (pow x_m 5.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m + ((-0.3333333333333333 * pow(x_m, 3.0)) + (0.13333333333333333 * pow(x_m, 5.0))));
}

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 + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (0.13333333333333333d0 * (x_m ** 5.0d0))))
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 + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (0.13333333333333333 * Math.pow(x_m, 5.0))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (0.13333333333333333 * math.pow(x_m, 5.0))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(0.13333333333333333 * (x_m ^ 5.0)))))
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 + ((-0.3333333333333333 * (x_m ^ 3.0)) + (0.13333333333333333 * (x_m ^ 5.0))));
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 * N[(x$95$m + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.2%

   \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

4. Final simplification 97.2%

   \[\leadsto x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right) \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + -0.3333333333333333 \cdot {x_m}^{3}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ x_m (* -0.3333333333333333 (pow x_m 3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m + (-0.3333333333333333 * pow(x_m, 3.0)));
}

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 + ((-0.3333333333333333d0) * (x_m ** 3.0d0)))
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 + (-0.3333333333333333 * Math.pow(x_m, 3.0)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m + (-0.3333333333333333 * math.pow(x_m, 3.0)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m + Float64(-0.3333333333333333 * (x_m ^ 3.0))))
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 + (-0.3333333333333333 * (x_m ^ 3.0)));
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 * N[(x$95$m + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 96.9%

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

   1. *-commutative 96.9%

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

5. Simplified 96.9%

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

6. Final simplification 96.9%

   \[\leadsto x + -0.3333333333333333 \cdot {x}^{3} \]
7. Add Preprocessing

Alternative 7: 5.8% accurate, 409.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot 1.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 1.5))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 1.5;
}

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 * 1.5d0
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 * 1.5;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 1.5

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 1.5)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 1.5;
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 * 1.5), $MachinePrecision]

Derivation

1. Initial program 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 8.5%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative 8.5%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{{x}^{2} + 2}} \]

   2. unpow2 8.5%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{x \cdot x} + 2} \]

   3. fma-def 8.5%

      \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Simplified 8.5%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

7. Applied egg-rr 4.3%

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

8. Taylor expanded in x around 0 4.4%

   \[\leadsto \color{blue}{1.5} \]

9. Final simplification 4.4%

   \[\leadsto 1.5 \]
10. Add Preprocessing

Alternative 8: 96.5% accurate, 409.0× 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 10.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 96.5%

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

4. Final simplification 96.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))