Hyperbolic tangent

Percentage Accurate: 9.4% → 99.2%

Time: 8.5s

Alternatives: 7

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: 9.4% 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: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{e^{x\_m} - t\_0}{e^{x\_m} + t\_0} \leq 0.01:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.13333333333333333 + {x\_m}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    x_s
    (if (<= (/ (- (exp x_m) t_0) (+ (exp x_m) t_0)) 0.01)
      (*
       x_m
       (+
        1.0
        (*
         (pow x_m 2.0)
         (-
          (*
           (pow x_m 2.0)
           (+ 0.13333333333333333 (* (pow x_m 2.0) -0.05396825396825397)))
          0.3333333333333333))))
      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)) <= 0.01) {
		tmp = x_m * (1.0 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.13333333333333333 + (pow(x_m, 2.0) * -0.05396825396825397))) - 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x_m)
    if (((exp(x_m) - t_0) / (exp(x_m) + t_0)) <= 0.01d0) then
        tmp = x_m * (1.0d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * (0.13333333333333333d0 + ((x_m ** 2.0d0) * (-0.05396825396825397d0)))) - 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 t_0 = Math.exp(-x_m);
	double tmp;
	if (((Math.exp(x_m) - t_0) / (Math.exp(x_m) + t_0)) <= 0.01) {
		tmp = x_m * (1.0 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.13333333333333333 + (Math.pow(x_m, 2.0) * -0.05396825396825397))) - 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):
	t_0 = math.exp(-x_m)
	tmp = 0
	if ((math.exp(x_m) - t_0) / (math.exp(x_m) + t_0)) <= 0.01:
		tmp = x_m * (1.0 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * (0.13333333333333333 + (math.pow(x_m, 2.0) * -0.05396825396825397))) - 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)
	t_0 = exp(Float64(-x_m))
	tmp = 0.0
	if (Float64(Float64(exp(x_m) - t_0) / Float64(exp(x_m) + t_0)) <= 0.01)
		tmp = Float64(x_m * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.13333333333333333 + Float64((x_m ^ 2.0) * -0.05396825396825397))) - 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)
	t_0 = exp(-x_m);
	tmp = 0.0;
	if (((exp(x_m) - t_0) / (exp(x_m) + t_0)) <= 0.01)
		tmp = x_m * (1.0 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * (0.13333333333333333 + ((x_m ^ 2.0) * -0.05396825396825397))) - 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_] := 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], 0.01], N[(x$95$m * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.13333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $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)))) < 0.0100000000000000002

   1. Initial program 8.6%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]

   if 0.0100000000000000002 < (/.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%

      \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

   6. Applied egg-rr 43.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 0.01:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left({x\_m}^{2} \cdot 0.13333333333333333 - 0.3333333333333333, {x\_m}^{3}, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.22)
    (fma
     (- (* (pow x_m 2.0) 0.13333333333333333) 0.3333333333333333)
     (pow x_m 3.0)
     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 <= 1.22) {
		tmp = fma(((pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333), pow(x_m, 3.0), 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 <= 1.22)
		tmp = fma(Float64(Float64((x_m ^ 2.0) * 0.13333333333333333) - 0.3333333333333333), (x_m ^ 3.0), x_m);
	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, 1.22], N[(N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[Power[x$95$m, 3.0], $MachinePrecision] + x$95$m), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 97.8%

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

      2. distribute-rgt-in 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*l* 97.8%

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

      5. *-lft-identity 97.8%

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

      6. fma-define 97.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333, {x}^{2} \cdot x, x\right)} \]

      7. *-commutative 97.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot 0.13333333333333333} - 0.3333333333333333, {x}^{2} \cdot x, x\right) \]

      8. fma-neg 97.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right)}, {x}^{2} \cdot x, x\right) \]

      9. metadata-eval 97.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, \color{blue}{-0.3333333333333333}\right), {x}^{2} \cdot x, x\right) \]

      10. pow-plus 97.8%

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

      11. metadata-eval 97.8%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right), {x}^{\color{blue}{3}}, x\right) \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right), {x}^{3}, x\right)} \]

   6. Taylor expanded in x around 0 97.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333}, {x}^{3}, x\right) \]

   if 1.21999999999999997 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

      \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333, {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.22:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.22)
    (*
     x_m
     (+
      1.0
      (*
       (pow x_m 2.0)
       (- (* (pow x_m 2.0) 0.13333333333333333) 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 <= 1.22) {
		tmp = x_m * (1.0 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.13333333333333333) - 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 <= 1.22d0) then
        tmp = x_m * (1.0d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.13333333333333333d0) - 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 <= 1.22) {
		tmp = x_m * (1.0 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.13333333333333333) - 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 <= 1.22:
		tmp = x_m * (1.0 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.13333333333333333) - 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 <= 1.22)
		tmp = Float64(x_m * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.13333333333333333) - 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 <= 1.22)
		tmp = x_m * (1.0 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.13333333333333333) - 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, 1.22], N[(x$95$m * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0 97.8%

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

   if 1.21999999999999997 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

      \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.6)
    (/ (* x_m (+ 2.0 (* (pow x_m 2.0) 0.3333333333333333))) (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 + (pow(x_m, 2.0) * 0.3333333333333333))) / 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((x_m ^ 2.0) * 0.3333333333333333))) / 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[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.3333333333333333), $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 8.5%

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

   3. Taylor expanded in x around 0 97.6%

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

   4. Taylor expanded in x around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. unpow2 97.8%

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

      3. fma-define 97.8%

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

   6. Simplified 97.8%

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

   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%

      \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\frac{x \cdot \left(2 + {x}^{2} \cdot 0.3333333333333333\right)}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.15) (+ 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 <= 1.15) {
		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 <= 1.15d0) 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 <= 1.15) {
		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 <= 1.15:
		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 <= 1.15)
		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 <= 1.15)
		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, 1.15], 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 < 1.1499999999999999

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 97.8%

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

      2. distribute-rgt-in 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*l* 97.8%

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

      5. *-lft-identity 97.8%

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

      6. fma-define 97.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333, {x}^{2} \cdot x, x\right)} \]

      7. *-commutative 97.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot 0.13333333333333333} - 0.3333333333333333, {x}^{2} \cdot x, x\right) \]

      8. fma-neg 97.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right)}, {x}^{2} \cdot x, x\right) \]

      9. metadata-eval 97.8%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, \color{blue}{-0.3333333333333333}\right), {x}^{2} \cdot x, x\right) \]

      10. pow-plus 97.8%

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

      11. metadata-eval 97.8%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right), {x}^{\color{blue}{3}}, x\right) \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right), {x}^{3}, x\right)} \]

   6. Taylor expanded in x around 0 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. *-lft-identity 97.4%

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

      3. associate-*l* 97.4%

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

      4. unpow2 97.4%

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

      5. unpow3 97.4%

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

   8. Simplified 97.4%

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

   if 1.1499999999999999 < x

   1. Initial program 0.0%

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

   4. Applied egg-rr 3.1%

      \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x + {x}^{3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% 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 1 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 8.5%

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

   3. Taylor expanded in x around 0 97.4%

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

   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%

      \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 8.0% 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 1 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 8.4%

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

4. Applied egg-rr 3.7%

   \[\leadsto \frac{\color{blue}{4}}{e^{x} + e^{-x}} \]

6. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 4.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))