Hyperbolic tangent

Percentage Accurate: 9.2% → 98.1%

Time: 12.1s

Alternatives: 6

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: 9.2% 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: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x\_m}\\ t_1 := \frac{e^{x\_m} - t\_0}{e^{x\_m} + t\_0}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0.005:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.13333333333333333 + {x\_m}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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))) (t_1 (/ (- (exp x_m) t_0) (+ (exp x_m) t_0))))
   (*
    x_s
    (if (<= t_1 0.005)
      (*
       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))))
      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 <= 0.005) {
		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 = 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 <= 0.005d0) 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 = 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 <= 0.005) {
		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 = 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 <= 0.005:
		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 = 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 <= 0.005)
		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 = 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 <= 0.005)
		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 = 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, 0.005], 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], 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)))) < 0.0050000000000000001

   1. Initial program 9.5%

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

   3. Taylor expanded in x around 0 97.8%

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

   if 0.0050000000000000001 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 39.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 1.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   x_m
   (+
    1.0
    (*
     (pow x_m 2.0)
     (- (* (pow x_m 2.0) 0.13333333333333333) 0.3333333333333333))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333))));
}

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 * (1.0d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.13333333333333333d0) - 0.3333333333333333d0))))
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 * (1.0 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * (1.0 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.13333333333333333) - 0.3333333333333333))))

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(1.0 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.13333333333333333) - 0.3333333333333333)))))
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 * (1.0 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.13333333333333333) - 0.3333333333333333))));
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[(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]), $MachinePrecision]

Derivation

1. Initial program 10.1%

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

3. Taylor expanded in x around 0 96.5%

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

4. Final simplification 96.5%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right) \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{x\_m}{2} \cdot \frac{1 + \frac{\mathsf{expm1}\left(x\_m\right)}{x\_m}}{\cosh x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (/ x_m 2.0) (/ (+ 1.0 (/ (expm1 x_m) x_m)) (cosh 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 / 2.0) * ((1.0 + (expm1(x_m) / x_m)) / cosh(x_m)));
}

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 / 2.0) * ((1.0 + (Math.expm1(x_m) / x_m)) / Math.cosh(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 / 2.0) * ((1.0 + (math.expm1(x_m) / x_m)) / math.cosh(x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m / 2.0) * Float64(Float64(1.0 + Float64(expm1(x_m) / x_m)) / cosh(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 * N[(N[(x$95$m / 2.0), $MachinePrecision] * N[(N[(1.0 + N[(N[(Exp[x$95$m] - 1), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.1%

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

3. Taylor expanded in x around 0 7.2%

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

   1. mul-1-neg 7.2%

      \[\leadsto \frac{e^{x} - \left(1 + \color{blue}{\left(-x\right)}\right)}{e^{x} + e^{-x}} \]

   2. unsub-neg 7.2%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{e^{x} + e^{-x}} \]

5. Simplified 7.2%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{e^{x} + e^{-x}} \]

6. Taylor expanded in x around inf 8.2%

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

   1. associate--l+ 20.4%

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

   2. div-sub 19.4%

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

   3. +-commutative 19.4%

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

   4. remove-double-neg 19.4%

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

   5. mul-1-neg 19.4%

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

   6. neg-sub0 19.4%

      \[\leadsto \frac{x \cdot \left(\color{blue}{\left(0 - -1 \cdot \frac{e^{x} - 1}{x}\right)} + 1\right)}{e^{x} + e^{-x}} \]

   7. associate--r- 19.4%

      \[\leadsto \frac{x \cdot \color{blue}{\left(0 - \left(-1 \cdot \frac{e^{x} - 1}{x} - 1\right)\right)}}{e^{x} + e^{-x}} \]

   8. neg-sub0 19.4%

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

   9. neg-sub0 19.4%

      \[\leadsto \frac{x \cdot \color{blue}{\left(0 - \left(-1 \cdot \frac{e^{x} - 1}{x} - 1\right)\right)}}{e^{x} + e^{-x}} \]

   10. associate--r- 19.4%

       \[\leadsto \frac{x \cdot \color{blue}{\left(\left(0 - -1 \cdot \frac{e^{x} - 1}{x}\right) + 1\right)}}{e^{x} + e^{-x}} \]

   11. neg-sub0 19.4%

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

   12. mul-1-neg 19.4%

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

   13. remove-double-neg 19.4%

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

   14. +-commutative 19.4%

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

   15. expm1-define 95.3%

       \[\leadsto \frac{x \cdot \left(1 + \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{x}\right)}{e^{x} + e^{-x}} \]

8. Simplified 95.3%

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + \frac{\mathsf{expm1}\left(x\right)}{x}\right)}}{e^{x} + e^{-x}} \]
9. Step-by-step derivation

   1. cosh-undef 95.3%

      \[\leadsto \frac{x \cdot \left(1 + \frac{\mathsf{expm1}\left(x\right)}{x}\right)}{\color{blue}{2 \cdot \cosh x}} \]

   2. times-frac 95.3%

      \[\leadsto \color{blue}{\frac{x}{2} \cdot \frac{1 + \frac{\mathsf{expm1}\left(x\right)}{x}}{\cosh x}} \]

10. Applied egg-rr 95.3%

    \[\leadsto \color{blue}{\frac{x}{2} \cdot \frac{1 + \frac{\mathsf{expm1}\left(x\right)}{x}}{\cosh x}} \]
11. Add Preprocessing

Alternative 4: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(x\_m + \mathsf{expm1}\left(x\_m\right)\right) \cdot \frac{0.5}{\cosh x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (+ x_m (expm1 x_m)) (/ 0.5 (cosh 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 + expm1(x_m)) * (0.5 / cosh(x_m)));
}

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 + Math.expm1(x_m)) * (0.5 / Math.cosh(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 + math.expm1(x_m)) * (0.5 / math.cosh(x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m + expm1(x_m)) * Float64(0.5 / cosh(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 * N[(N[(x$95$m + N[(Exp[x$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.1%

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

3. Taylor expanded in x around 0 7.2%

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

   1. mul-1-neg 7.2%

      \[\leadsto \frac{e^{x} - \left(1 + \color{blue}{\left(-x\right)}\right)}{e^{x} + e^{-x}} \]

   2. unsub-neg 7.2%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{e^{x} + e^{-x}} \]

5. Simplified 7.2%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{e^{x} + e^{-x}} \]

6. Taylor expanded in x around inf 8.2%

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

   1. associate--l+ 20.4%

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

   2. div-sub 19.4%

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

   3. distribute-rgt-in 19.4%

      \[\leadsto \frac{\color{blue}{1 \cdot x + \frac{e^{x} - 1}{x} \cdot x}}{e^{x} + e^{-x}} \]

   4. *-lft-identity 19.4%

      \[\leadsto \frac{\color{blue}{x} + \frac{e^{x} - 1}{x} \cdot x}{e^{x} + e^{-x}} \]

   5. *-lft-identity 19.4%

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

   6. metadata-eval 19.4%

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

   7. associate-*r* 19.4%

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

   8. associate-*l* 19.4%

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

   9. *-commutative 19.4%

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

   10. *-commutative 19.4%

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

   11. neg-mul-1 19.4%

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

   12. distribute-lft-neg-out 19.4%

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

   13. unsub-neg 19.4%

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

   14. *-commutative 19.4%

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

   15. mul-1-neg 19.4%

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

   16. distribute-neg-frac2 19.4%

       \[\leadsto \frac{x - \color{blue}{\frac{e^{x} - 1}{-x}} \cdot x}{e^{x} + e^{-x}} \]

   17. neg-mul-1 19.4%

       \[\leadsto \frac{x - \frac{e^{x} - 1}{\color{blue}{-1 \cdot x}} \cdot x}{e^{x} + e^{-x}} \]

   18. associate-*l/ 19.4%

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

8. Simplified 95.3%

   \[\leadsto \frac{\color{blue}{x - \left(-\mathsf{expm1}\left(x\right)\right)}}{e^{x} + e^{-x}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 95.3%

      \[\leadsto \color{blue}{1 \cdot \frac{x - \left(-\mathsf{expm1}\left(x\right)\right)}{e^{x} + e^{-x}}} \]

   2. *-un-lft-identity 95.3%

      \[\leadsto 1 \cdot \frac{\color{blue}{1 \cdot \left(x - \left(-\mathsf{expm1}\left(x\right)\right)\right)}}{e^{x} + e^{-x}} \]

   3. cosh-undef 95.3%

      \[\leadsto 1 \cdot \frac{1 \cdot \left(x - \left(-\mathsf{expm1}\left(x\right)\right)\right)}{\color{blue}{2 \cdot \cosh x}} \]

   4. times-frac 95.3%

      \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{x - \left(-\mathsf{expm1}\left(x\right)\right)}{\cosh x}\right)} \]

   5. metadata-eval 95.3%

      \[\leadsto 1 \cdot \left(\color{blue}{0.5} \cdot \frac{x - \left(-\mathsf{expm1}\left(x\right)\right)}{\cosh x}\right) \]

   6. sub-neg 95.3%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{\color{blue}{x + \left(-\left(-\mathsf{expm1}\left(x\right)\right)\right)}}{\cosh x}\right) \]

   7. add-sqr-sqrt 48.2%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \left(-\color{blue}{\sqrt{-\mathsf{expm1}\left(x\right)} \cdot \sqrt{-\mathsf{expm1}\left(x\right)}}\right)}{\cosh x}\right) \]

   8. sqrt-unprod 34.0%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \left(-\color{blue}{\sqrt{\left(-\mathsf{expm1}\left(x\right)\right) \cdot \left(-\mathsf{expm1}\left(x\right)\right)}}\right)}{\cosh x}\right) \]

   9. sqr-neg 34.0%

      \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \left(-\sqrt{\color{blue}{\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)}}\right)}{\cosh x}\right) \]

   10. sqrt-unprod 2.9%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \left(-\color{blue}{\sqrt{\mathsf{expm1}\left(x\right)} \cdot \sqrt{\mathsf{expm1}\left(x\right)}}\right)}{\cosh x}\right) \]

   11. add-sqr-sqrt 5.8%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \left(-\color{blue}{\mathsf{expm1}\left(x\right)}\right)}{\cosh x}\right) \]

   12. add-sqr-sqrt 3.5%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \color{blue}{\sqrt{-\mathsf{expm1}\left(x\right)} \cdot \sqrt{-\mathsf{expm1}\left(x\right)}}}{\cosh x}\right) \]

   13. sqrt-unprod 33.0%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \color{blue}{\sqrt{\left(-\mathsf{expm1}\left(x\right)\right) \cdot \left(-\mathsf{expm1}\left(x\right)\right)}}}{\cosh x}\right) \]

   14. sqr-neg 33.0%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \sqrt{\color{blue}{\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)}}}{\cosh x}\right) \]

   15. sqrt-unprod 46.8%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \color{blue}{\sqrt{\mathsf{expm1}\left(x\right)} \cdot \sqrt{\mathsf{expm1}\left(x\right)}}}{\cosh x}\right) \]

   16. add-sqr-sqrt 95.3%

       \[\leadsto 1 \cdot \left(0.5 \cdot \frac{x + \color{blue}{\mathsf{expm1}\left(x\right)}}{\cosh x}\right) \]

10. Applied egg-rr 95.3%

    \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{\cosh x}\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 95.3%

       \[\leadsto \color{blue}{0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{\cosh x}} \]

    2. metadata-eval 95.3%

       \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{x + \mathsf{expm1}\left(x\right)}{\cosh x} \]

    3. times-frac 95.3%

       \[\leadsto \color{blue}{\frac{1 \cdot \left(x + \mathsf{expm1}\left(x\right)\right)}{2 \cdot \cosh x}} \]

    4. *-commutative 95.3%

       \[\leadsto \frac{\color{blue}{\left(x + \mathsf{expm1}\left(x\right)\right) \cdot 1}}{2 \cdot \cosh x} \]

    5. associate-*r/ 95.3%

       \[\leadsto \color{blue}{\left(x + \mathsf{expm1}\left(x\right)\right) \cdot \frac{1}{2 \cdot \cosh x}} \]

    6. associate-/r* 95.3%

       \[\leadsto \left(x + \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{\cosh x}} \]

    7. metadata-eval 95.3%

       \[\leadsto \left(x + \mathsf{expm1}\left(x\right)\right) \cdot \frac{\color{blue}{0.5}}{\cosh x} \]

12. Simplified 95.3%

    \[\leadsto \color{blue}{\left(x + \mathsf{expm1}\left(x\right)\right) \cdot \frac{0.5}{\cosh x}} \]
13. Add Preprocessing

Alternative 5: 96.7% 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 #s(literal 1 binary64) 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 10.1%

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

3. Taylor expanded in x around 0 96.1%

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

Alternative 6: 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 #s(literal 1 binary64) 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 10.1%

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

4. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{1.5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))