Hyperbolic tangent

Percentage Accurate: 9.6% → 98.7%

Time: 10.4s

Alternatives: 6

Speedup: 68.0×

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: 9.6% 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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{e^{x\_m} - t\_0}{e^{x\_m} + t\_0} \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 2, 0.3333333333333333 \cdot {x\_m}^{3}\right)}{e^{x\_m} + \left(1 + x\_m \cdot \left(x\_m \cdot \left(0.5 + x\_m \cdot -0.16666666666666666\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))))
   (*
    x_s
    (if (<= (/ (- (exp x_m) t_0) (+ (exp x_m) t_0)) 1.0)
      (/
       (fma x_m 2.0 (* 0.3333333333333333 (pow x_m 3.0)))
       (+
        (exp x_m)
        (+ 1.0 (* x_m (+ (* x_m (+ 0.5 (* x_m -0.16666666666666666))) -1.0)))))
      1.0))))

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 tmp;
	if (((exp(x_m) - t_0) / (exp(x_m) + t_0)) <= 1.0) {
		tmp = fma(x_m, 2.0, (0.3333333333333333 * pow(x_m, 3.0))) / (exp(x_m) + (1.0 + (x_m * ((x_m * (0.5 + (x_m * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = 1.0;
	}
	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))
	tmp = 0.0
	if (Float64(Float64(exp(x_m) - t_0) / Float64(exp(x_m) + t_0)) <= 1.0)
		tmp = Float64(fma(x_m, 2.0, Float64(0.3333333333333333 * (x_m ^ 3.0))) / Float64(exp(x_m) + Float64(1.0 + Float64(x_m * Float64(Float64(x_m * Float64(0.5 + Float64(x_m * -0.16666666666666666))) + -1.0)))));
	else
		tmp = 1.0;
	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_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[Exp[x$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(x$95$m * 2.0 + N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + N[(1.0 + N[(x$95$m * N[(N[(x$95$m * N[(0.5 + N[(x$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $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)))) < 1

   1. Initial program 7.2%

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

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{e^{x} + \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)}} \]

   5. Taylor expanded in x around inf 38.7%

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(0.3333333333333333 + 2 \cdot \frac{1}{{x}^{2}}\right)}}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 38.7%

         \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 0.3333333333333333\right)}}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      2. distribute-lft-in 38.7%

         \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \left(2 \cdot \frac{1}{{x}^{2}}\right) + {x}^{3} \cdot 0.3333333333333333}}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      3. cube-mult 38.8%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(2 \cdot \frac{1}{{x}^{2}}\right) + {x}^{3} \cdot 0.3333333333333333}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      4. unpow2 38.8%

         \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(2 \cdot \frac{1}{{x}^{2}}\right) + {x}^{3} \cdot 0.3333333333333333}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      5. associate-*l* 53.8%

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)} + {x}^{3} \cdot 0.3333333333333333}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      6. *-commutative 53.8%

         \[\leadsto \frac{x \cdot \left({x}^{2} \cdot \left(2 \cdot \frac{1}{{x}^{2}}\right)\right) + \color{blue}{0.3333333333333333 \cdot {x}^{3}}}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      7. fma-define 53.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left(2 \cdot \frac{1}{{x}^{2}}\right), 0.3333333333333333 \cdot {x}^{3}\right)}}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      8. *-commutative 53.8%

         \[\leadsto \frac{\mathsf{fma}\left(x, {x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot 2\right)}, 0.3333333333333333 \cdot {x}^{3}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      9. associate-*r* 53.8%

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right) \cdot 2}, 0.3333333333333333 \cdot {x}^{3}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      10. rgt-mult-inverse 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{1} \cdot 2, 0.3333333333333333 \cdot {x}^{3}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

      11. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{2}, 0.3333333333333333 \cdot {x}^{3}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 2, 0.3333333333333333 \cdot {x}^{3}\right)}}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

   if 1 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 3.6%

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

      1. +-commutative 3.6%

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

      2. unpow2 3.6%

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

      3. fma-define 3.6%

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

   7. Simplified 3.6%

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

   8. Taylor expanded in x around 0 23.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 2, 0.3333333333333333 \cdot {x}^{3}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right)}{e^{x\_m} + \left(1 + x\_m \cdot \left(x\_m \cdot \left(0.5 + x\_m \cdot -0.16666666666666666\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1)
    (/
     (* x_m (+ 2.0 (* 0.3333333333333333 (* x_m x_m))))
     (+
      (exp x_m)
      (+ 1.0 (* x_m (+ (* x_m (+ 0.5 (* x_m -0.16666666666666666))) -1.0)))))
    1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.1) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / (exp(x_m) + (1.0 + (x_m * ((x_m * (0.5 + (x_m * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = 1.0;
	}
	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 <= 2.1d0) then
        tmp = (x_m * (2.0d0 + (0.3333333333333333d0 * (x_m * x_m)))) / (exp(x_m) + (1.0d0 + (x_m * ((x_m * (0.5d0 + (x_m * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = 1.0d0
    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 <= 2.1) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / (Math.exp(x_m) + (1.0 + (x_m * ((x_m * (0.5 + (x_m * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = 1.0;
	}
	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 <= 2.1:
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / (math.exp(x_m) + (1.0 + (x_m * ((x_m * (0.5 + (x_m * -0.16666666666666666))) + -1.0))))
	else:
		tmp = 1.0
	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 <= 2.1)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * Float64(x_m * x_m)))) / Float64(exp(x_m) + Float64(1.0 + Float64(x_m * Float64(Float64(x_m * Float64(0.5 + Float64(x_m * -0.16666666666666666))) + -1.0)))));
	else
		tmp = 1.0;
	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 <= 2.1)
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / (exp(x_m) + (1.0 + (x_m * ((x_m * (0.5 + (x_m * -0.16666666666666666))) + -1.0))));
	else
		tmp = 1.0;
	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, 2.1], N[(N[(x$95$m * N[(2.0 + N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + N[(1.0 + N[(x$95$m * N[(N[(x$95$m * N[(0.5 + N[(x$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 97.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 97.5%

      \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{e^{x} + \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 97.5%

         \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

   6. Applied egg-rr 97.5%

      \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

   if 2.10000000000000009 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 10.7%

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

      1. +-commutative 10.7%

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

      2. unpow2 10.7%

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

      3. fma-define 10.7%

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

   7. Simplified 10.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right)}{\mathsf{fma}\left(x\_m, x\_m, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.6)
    (/ (* x_m (+ 2.0 (* 0.3333333333333333 (* x_m x_m)))) (fma x_m x_m 2.0))
    1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.6) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / fma(x_m, x_m, 2.0);
	} else {
		tmp = 1.0;
	}
	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 <= 1.6)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * Float64(x_m * x_m)))) / fma(x_m, x_m, 2.0));
	else
		tmp = 1.0;
	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, 1.6], N[(N[(x$95$m * N[(2.0 + N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 97.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 97.3%

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

      1. +-commutative 3.8%

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

      2. unpow2 3.8%

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

      3. fma-define 3.8%

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

   6. Simplified 97.3%

      \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
   7. Step-by-step derivation

      1. unpow2 97.5%

         \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{e^{x} + \left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \]

   8. Applied egg-rr 97.3%

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

   if 1.6000000000000001 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 10.7%

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

      1. +-commutative 10.7%

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

      2. unpow2 10.7%

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

      3. fma-define 10.7%

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

   7. Simplified 10.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.16:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.16) (+ x_m (* (pow x_m 3.0) -0.3333333333333333)) 1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.16) {
		tmp = x_m + (pow(x_m, 3.0) * -0.3333333333333333);
	} else {
		tmp = 1.0;
	}
	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 <= 1.16d0) then
        tmp = x_m + ((x_m ** 3.0d0) * (-0.3333333333333333d0))
    else
        tmp = 1.0d0
    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 <= 1.16) {
		tmp = x_m + (Math.pow(x_m, 3.0) * -0.3333333333333333);
	} else {
		tmp = 1.0;
	}
	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 <= 1.16:
		tmp = x_m + (math.pow(x_m, 3.0) * -0.3333333333333333)
	else:
		tmp = 1.0
	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 <= 1.16)
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * -0.3333333333333333));
	else
		tmp = 1.0;
	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 <= 1.16)
		tmp = x_m + ((x_m ^ 3.0) * -0.3333333333333333);
	else
		tmp = 1.0;
	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, 1.16], N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.15999999999999992

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 97.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]

   4. Taylor expanded in x around 0 97.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 97.0%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 97.0%

         \[\leadsto \color{blue}{x} + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*r* 97.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 97.0%

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

      5. unpow3 97.0%

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

   6. Simplified 97.0%

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

   if 1.15999999999999992 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 10.7%

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

      1. +-commutative 10.7%

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

      2. unpow2 10.7%

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

      3. fma-define 10.7%

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

   7. Simplified 10.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16:\\ \;\;\;\;x + {x}^{3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 68.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (if (<= x_m 1.0) x_m 1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0;
	}
	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 <= 1.0d0) then
        tmp = x_m
    else
        tmp = 1.0d0
    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 <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0;
	}
	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 <= 1.0:
		tmp = x_m
	else:
		tmp = 1.0
	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 <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0;
	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 <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0;
	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, 1.0], x$95$m, 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 97.1%

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

   if 1 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 10.7%

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

      1. +-commutative 10.7%

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

      2. unpow2 10.7%

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

      3. fma-define 10.7%

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

   7. Simplified 10.7%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 7.8% accurate, 409.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 1.0))

C

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

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 * 1.0d0
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 * 1.0;
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 7.0%

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

4. Applied egg-rr 3.8%

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

5. Taylor expanded in x around 0 3.8%

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

   1. +-commutative 3.8%

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

   2. unpow2 3.8%

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

   3. fma-define 3.8%

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

7. Simplified 3.8%

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

8. Taylor expanded in x around 0 4.5%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))