Hyperbolic sine

Percentage Accurate: 53.2% → 100.0%

Time: 9.2s

Alternatives: 8

Speedup: 1.9×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\left(-0.016666666666666666 \cdot {x\_m}^{5} + -0.3333333333333333 \cdot {x\_m}^{3}\right) + \left(-0.0003968253968253968 \cdot {x\_m}^{7} + x\_m \cdot -2\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.1)
      (/
       (+
        (+
         (* -0.016666666666666666 (pow x_m 5.0))
         (* -0.3333333333333333 (pow x_m 3.0)))
        (+ (* -0.0003968253968253968 (pow x_m 7.0)) (* x_m -2.0)))
       -2.0)
      (/ t_0 2.0)))))

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) - exp(-x_m);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = (((-0.016666666666666666 * pow(x_m, 5.0)) + (-0.3333333333333333 * pow(x_m, 3.0))) + ((-0.0003968253968253968 * pow(x_m, 7.0)) + (x_m * -2.0))) / -2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	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) - exp(-x_m)
    if (t_0 <= 0.1d0) then
        tmp = ((((-0.016666666666666666d0) * (x_m ** 5.0d0)) + ((-0.3333333333333333d0) * (x_m ** 3.0d0))) + (((-0.0003968253968253968d0) * (x_m ** 7.0d0)) + (x_m * (-2.0d0)))) / (-2.0d0)
    else
        tmp = t_0 / 2.0d0
    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) - Math.exp(-x_m);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = (((-0.016666666666666666 * Math.pow(x_m, 5.0)) + (-0.3333333333333333 * Math.pow(x_m, 3.0))) + ((-0.0003968253968253968 * Math.pow(x_m, 7.0)) + (x_m * -2.0))) / -2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	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) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 0.1:
		tmp = (((-0.016666666666666666 * math.pow(x_m, 5.0)) + (-0.3333333333333333 * math.pow(x_m, 3.0))) + ((-0.0003968253968253968 * math.pow(x_m, 7.0)) + (x_m * -2.0))) / -2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(Float64(Float64(Float64(-0.016666666666666666 * (x_m ^ 5.0)) + Float64(-0.3333333333333333 * (x_m ^ 3.0))) + Float64(Float64(-0.0003968253968253968 * (x_m ^ 7.0)) + Float64(x_m * -2.0))) / -2.0);
	else
		tmp = Float64(t_0 / 2.0);
	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) - exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = (((-0.016666666666666666 * (x_m ^ 5.0)) + (-0.3333333333333333 * (x_m ^ 3.0))) + ((-0.0003968253968253968 * (x_m ^ 7.0)) + (x_m * -2.0))) / -2.0;
	else
		tmp = t_0 / 2.0;
	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[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.1], N[(N[(N[(N[(-0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.10000000000000001

   1. Initial program 36.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 36.5%

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

      2. remove-double-neg 36.5%

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

      3. remove-double-neg 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. +-commutative 36.5%

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

      6. sub-neg 36.5%

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

      7. distribute-neg-frac 36.5%

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

      8. distribute-neg-frac2 36.5%

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

      9. remove-double-neg 36.5%

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

      10. metadata-eval 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}}{-2} \]
   6. Step-by-step derivation

      1. associate-+r+ 97.3%

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)}}{-2} \]

      2. +-commutative 97.3%

         \[\leadsto \frac{\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + \color{blue}{\left(-0.0003968253968253968 \cdot {x}^{7} + -0.016666666666666666 \cdot {x}^{5}\right)}}{-2} \]

      3. associate-+r+ 97.3%

         \[\leadsto \frac{\color{blue}{\left(\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + -0.0003968253968253968 \cdot {x}^{7}\right) + -0.016666666666666666 \cdot {x}^{5}}}{-2} \]

      4. +-commutative 97.3%

         \[\leadsto \frac{\color{blue}{-0.016666666666666666 \cdot {x}^{5} + \left(\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + -0.0003968253968253968 \cdot {x}^{7}\right)}}{-2} \]

      5. fma-define 97.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + -0.0003968253968253968 \cdot {x}^{7}\right)}}{-2} \]

      6. associate-+l+ 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + -0.0003968253968253968 \cdot {x}^{7}\right)}\right)}{-2} \]

      7. fma-define 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \color{blue}{\mathsf{fma}\left(-2, x, -0.3333333333333333 \cdot {x}^{3} + -0.0003968253968253968 \cdot {x}^{7}\right)}\right)}{-2} \]

      8. +-commutative 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \mathsf{fma}\left(-2, x, \color{blue}{-0.0003968253968253968 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}}\right)\right)}{-2} \]

      9. fma-define 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \mathsf{fma}\left(-2, x, \color{blue}{\mathsf{fma}\left(-0.0003968253968253968, {x}^{7}, -0.3333333333333333 \cdot {x}^{3}\right)}\right)\right)}{-2} \]

   7. Simplified 97.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \mathsf{fma}\left(-2, x, \mathsf{fma}\left(-0.0003968253968253968, {x}^{7}, -0.3333333333333333 \cdot {x}^{3}\right)\right)\right)}}{-2} \]
   8. Step-by-step derivation

      1. fma-undefine 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \mathsf{fma}\left(-2, x, \color{blue}{-0.0003968253968253968 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}}\right)\right)}{-2} \]

   9. Applied egg-rr 97.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.016666666666666666, {x}^{5}, \mathsf{fma}\left(-2, x, \color{blue}{-0.0003968253968253968 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}}\right)\right)}{-2} \]
   10. Step-by-step derivation

       1. fma-undefine 97.3%

          \[\leadsto \frac{\color{blue}{-0.016666666666666666 \cdot {x}^{5} + \mathsf{fma}\left(-2, x, -0.0003968253968253968 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}\right)}}{-2} \]

       2. +-commutative 97.3%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, x, -0.0003968253968253968 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}\right) + -0.016666666666666666 \cdot {x}^{5}}}{-2} \]

       3. fma-undefine 97.3%

          \[\leadsto \frac{\color{blue}{\left(-2 \cdot x + \left(-0.0003968253968253968 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}\right)\right)} + -0.016666666666666666 \cdot {x}^{5}}{-2} \]

       4. +-commutative 97.3%

          \[\leadsto \frac{\left(-2 \cdot x + \color{blue}{\left(-0.3333333333333333 \cdot {x}^{3} + -0.0003968253968253968 \cdot {x}^{7}\right)}\right) + -0.016666666666666666 \cdot {x}^{5}}{-2} \]

       5. associate-+r+ 97.3%

          \[\leadsto \frac{\color{blue}{\left(\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + -0.0003968253968253968 \cdot {x}^{7}\right)} + -0.016666666666666666 \cdot {x}^{5}}{-2} \]

       6. associate-+r+ 97.3%

          \[\leadsto \frac{\color{blue}{\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + \left(-0.0003968253968253968 \cdot {x}^{7} + -0.016666666666666666 \cdot {x}^{5}\right)}}{-2} \]

       7. +-commutative 97.3%

          \[\leadsto \frac{\left(-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}\right) + \color{blue}{\left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)}}{-2} \]

       8. associate-+r+ 97.3%

          \[\leadsto \frac{\color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}}{-2} \]

       9. +-commutative 97.3%

          \[\leadsto \frac{\color{blue}{\left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right) + -2 \cdot x}}{-2} \]

       10. associate-+r+ 97.3%

           \[\leadsto \frac{\color{blue}{\left(\left(-0.3333333333333333 \cdot {x}^{3} + -0.016666666666666666 \cdot {x}^{5}\right) + -0.0003968253968253968 \cdot {x}^{7}\right)} + -2 \cdot x}{-2} \]

       11. associate-+l+ 97.3%

           \[\leadsto \frac{\color{blue}{\left(-0.3333333333333333 \cdot {x}^{3} + -0.016666666666666666 \cdot {x}^{5}\right) + \left(-0.0003968253968253968 \cdot {x}^{7} + -2 \cdot x\right)}}{-2} \]

       12. fma-define 97.3%

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, -0.016666666666666666 \cdot {x}^{5}\right)} + \left(-0.0003968253968253968 \cdot {x}^{7} + -2 \cdot x\right)}{-2} \]

       13. *-commutative 97.3%

           \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, -0.016666666666666666 \cdot {x}^{5}\right) + \left(-0.0003968253968253968 \cdot {x}^{7} + \color{blue}{x \cdot -2}\right)}{-2} \]

   11. Applied egg-rr 97.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, -0.016666666666666666 \cdot {x}^{5}\right) + \left(-0.0003968253968253968 \cdot {x}^{7} + x \cdot -2\right)}}{-2} \]
   12. Step-by-step derivation

       1. fma-undefine 97.3%

          \[\leadsto \frac{\color{blue}{\left(-0.3333333333333333 \cdot {x}^{3} + -0.016666666666666666 \cdot {x}^{5}\right)} + \left(-0.0003968253968253968 \cdot {x}^{7} + x \cdot -2\right)}{-2} \]

       2. +-commutative 97.3%

          \[\leadsto \frac{\color{blue}{\left(-0.016666666666666666 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)} + \left(-0.0003968253968253968 \cdot {x}^{7} + x \cdot -2\right)}{-2} \]

   13. Applied egg-rr 97.3%

       \[\leadsto \frac{\color{blue}{\left(-0.016666666666666666 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)} + \left(-0.0003968253968253968 \cdot {x}^{7} + x \cdot -2\right)}{-2} \]

   if 0.10000000000000001 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\frac{\left(-0.016666666666666666 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right) + \left(-0.0003968253968253968 \cdot {x}^{7} + x \cdot -2\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{x\_m \cdot -2 + \left(-0.3333333333333333 \cdot {x\_m}^{3} + \left(-0.016666666666666666 \cdot {x\_m}^{5} + -0.0003968253968253968 \cdot {x\_m}^{7}\right)\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.1)
      (/
       (+
        (* x_m -2.0)
        (+
         (* -0.3333333333333333 (pow x_m 3.0))
         (+
          (* -0.016666666666666666 (pow x_m 5.0))
          (* -0.0003968253968253968 (pow x_m 7.0)))))
       -2.0)
      (/ t_0 2.0)))))

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) - exp(-x_m);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = ((x_m * -2.0) + ((-0.3333333333333333 * pow(x_m, 3.0)) + ((-0.016666666666666666 * pow(x_m, 5.0)) + (-0.0003968253968253968 * pow(x_m, 7.0))))) / -2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	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) - exp(-x_m)
    if (t_0 <= 0.1d0) then
        tmp = ((x_m * (-2.0d0)) + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (((-0.016666666666666666d0) * (x_m ** 5.0d0)) + ((-0.0003968253968253968d0) * (x_m ** 7.0d0))))) / (-2.0d0)
    else
        tmp = t_0 / 2.0d0
    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) - Math.exp(-x_m);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = ((x_m * -2.0) + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + ((-0.016666666666666666 * Math.pow(x_m, 5.0)) + (-0.0003968253968253968 * Math.pow(x_m, 7.0))))) / -2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	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) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 0.1:
		tmp = ((x_m * -2.0) + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + ((-0.016666666666666666 * math.pow(x_m, 5.0)) + (-0.0003968253968253968 * math.pow(x_m, 7.0))))) / -2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(Float64(Float64(x_m * -2.0) + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(Float64(-0.016666666666666666 * (x_m ^ 5.0)) + Float64(-0.0003968253968253968 * (x_m ^ 7.0))))) / -2.0);
	else
		tmp = Float64(t_0 / 2.0);
	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) - exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = ((x_m * -2.0) + ((-0.3333333333333333 * (x_m ^ 3.0)) + ((-0.016666666666666666 * (x_m ^ 5.0)) + (-0.0003968253968253968 * (x_m ^ 7.0))))) / -2.0;
	else
		tmp = t_0 / 2.0;
	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[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.1], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.10000000000000001

   1. Initial program 36.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 36.5%

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

      2. remove-double-neg 36.5%

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

      3. remove-double-neg 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. +-commutative 36.5%

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

      6. sub-neg 36.5%

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

      7. distribute-neg-frac 36.5%

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

      8. distribute-neg-frac2 36.5%

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

      9. remove-double-neg 36.5%

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

      10. metadata-eval 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}}{-2} \]

   if 0.10000000000000001 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\frac{x \cdot -2 + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x\_m}^{3} + x\_m \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 1e-7)
      (/ (+ (* -0.3333333333333333 (pow x_m 3.0)) (* x_m -2.0)) -2.0)
      (/ t_0 2.0)))))

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) - exp(-x_m);
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = ((-0.3333333333333333 * pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	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) - exp(-x_m)
    if (t_0 <= 1d-7) then
        tmp = (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (x_m * (-2.0d0))) / (-2.0d0)
    else
        tmp = t_0 / 2.0d0
    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) - Math.exp(-x_m);
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	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) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 1e-7:
		tmp = ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 1e-7)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * -2.0)) / -2.0);
	else
		tmp = Float64(t_0 / 2.0);
	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) - exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 1e-7)
		tmp = ((-0.3333333333333333 * (x_m ^ 3.0)) + (x_m * -2.0)) / -2.0;
	else
		tmp = t_0 / 2.0;
	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[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 1e-7], N[(N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 9.9999999999999995e-8

   1. Initial program 36.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 36.5%

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

      2. remove-double-neg 36.5%

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

      3. remove-double-neg 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. +-commutative 36.5%

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

      6. sub-neg 36.5%

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

      7. distribute-neg-frac 36.5%

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

      8. distribute-neg-frac2 36.5%

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

      9. remove-double-neg 36.5%

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

      10. metadata-eval 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 92.3%

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

   if 9.9999999999999995e-8 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 10^{-7}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x}^{3} + x \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.5:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x\_m}^{3} + x\_m \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x\_m}^{14} \cdot 3.936759889140842 \cdot 10^{-8}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.5)
    (/ (+ (* -0.3333333333333333 (pow x_m 3.0)) (* x_m -2.0)) -2.0)
    (sqrt (* (pow x_m 14.0) 3.936759889140842e-8)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.5) {
		tmp = ((-0.3333333333333333 * pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0;
	} else {
		tmp = sqrt((pow(x_m, 14.0) * 3.936759889140842e-8));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5.5d0) then
        tmp = (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (x_m * (-2.0d0))) / (-2.0d0)
    else
        tmp = sqrt(((x_m ** 14.0d0) * 3.936759889140842d-8))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.5) {
		tmp = ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0;
	} else {
		tmp = Math.sqrt((Math.pow(x_m, 14.0) * 3.936759889140842e-8));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5.5:
		tmp = ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0
	else:
		tmp = math.sqrt((math.pow(x_m, 14.0) * 3.936759889140842e-8))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5.5)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * -2.0)) / -2.0);
	else
		tmp = sqrt(Float64((x_m ^ 14.0) * 3.936759889140842e-8));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5.5)
		tmp = ((-0.3333333333333333 * (x_m ^ 3.0)) + (x_m * -2.0)) / -2.0;
	else
		tmp = sqrt(((x_m ^ 14.0) * 3.936759889140842e-8));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5.5], N[(N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[Sqrt[N[(N[Power[x$95$m, 14.0], $MachinePrecision] * 3.936759889140842e-8), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.5

   1. Initial program 36.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 36.5%

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

      2. remove-double-neg 36.5%

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

      3. remove-double-neg 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. +-commutative 36.5%

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

      6. sub-neg 36.5%

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

      7. distribute-neg-frac 36.5%

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

      8. distribute-neg-frac2 36.5%

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

      9. remove-double-neg 36.5%

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

      10. metadata-eval 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 92.3%

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

   if 5.5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. +-commutative 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. distribute-neg-frac2 100.0%

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

      9. remove-double-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}}{-2} \]

   6. Taylor expanded in x around inf 95.1%

      \[\leadsto \frac{\color{blue}{-0.0003968253968253968 \cdot {x}^{7}}}{-2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 95.1%

         \[\leadsto \color{blue}{\sqrt{\frac{-0.0003968253968253968 \cdot {x}^{7}}{-2}} \cdot \sqrt{\frac{-0.0003968253968253968 \cdot {x}^{7}}{-2}}} \]

      2. sqrt-unprod 96.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-0.0003968253968253968 \cdot {x}^{7}}{-2} \cdot \frac{-0.0003968253968253968 \cdot {x}^{7}}{-2}}} \]

      3. div-inv 96.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(-0.0003968253968253968 \cdot {x}^{7}\right) \cdot \frac{1}{-2}\right)} \cdot \frac{-0.0003968253968253968 \cdot {x}^{7}}{-2}} \]

      4. div-inv 96.7%

         \[\leadsto \sqrt{\left(\left(-0.0003968253968253968 \cdot {x}^{7}\right) \cdot \frac{1}{-2}\right) \cdot \color{blue}{\left(\left(-0.0003968253968253968 \cdot {x}^{7}\right) \cdot \frac{1}{-2}\right)}} \]

      5. swap-sqr 96.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(-0.0003968253968253968 \cdot {x}^{7}\right) \cdot \left(-0.0003968253968253968 \cdot {x}^{7}\right)\right) \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)}} \]

      6. *-commutative 96.7%

         \[\leadsto \sqrt{\left(\color{blue}{\left({x}^{7} \cdot -0.0003968253968253968\right)} \cdot \left(-0.0003968253968253968 \cdot {x}^{7}\right)\right) \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)} \]

      7. *-commutative 96.7%

         \[\leadsto \sqrt{\left(\left({x}^{7} \cdot -0.0003968253968253968\right) \cdot \color{blue}{\left({x}^{7} \cdot -0.0003968253968253968\right)}\right) \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)} \]

      8. swap-sqr 96.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left({x}^{7} \cdot {x}^{7}\right) \cdot \left(-0.0003968253968253968 \cdot -0.0003968253968253968\right)\right)} \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)} \]

      9. pow-prod-up 96.7%

         \[\leadsto \sqrt{\left(\color{blue}{{x}^{\left(7 + 7\right)}} \cdot \left(-0.0003968253968253968 \cdot -0.0003968253968253968\right)\right) \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)} \]

      10. metadata-eval 96.7%

          \[\leadsto \sqrt{\left({x}^{\color{blue}{14}} \cdot \left(-0.0003968253968253968 \cdot -0.0003968253968253968\right)\right) \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)} \]

      11. metadata-eval 96.7%

          \[\leadsto \sqrt{\left({x}^{14} \cdot \color{blue}{1.5747039556563367 \cdot 10^{-7}}\right) \cdot \left(\frac{1}{-2} \cdot \frac{1}{-2}\right)} \]

      12. metadata-eval 96.7%

          \[\leadsto \sqrt{\left({x}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}\right) \cdot \left(\color{blue}{-0.5} \cdot \frac{1}{-2}\right)} \]

      13. metadata-eval 96.7%

          \[\leadsto \sqrt{\left({x}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}\right) \cdot \left(-0.5 \cdot \color{blue}{-0.5}\right)} \]

      14. metadata-eval 96.7%

          \[\leadsto \sqrt{\left({x}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}\right) \cdot \color{blue}{0.25}} \]

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt{\left({x}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}\right) \cdot 0.25}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

         \[\leadsto \sqrt{\color{blue}{{x}^{14} \cdot \left(1.5747039556563367 \cdot 10^{-7} \cdot 0.25\right)}} \]

      2. metadata-eval 96.7%

         \[\leadsto \sqrt{{x}^{14} \cdot \color{blue}{3.936759889140842 \cdot 10^{-8}}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{14} \cdot 3.936759889140842 \cdot 10^{-8}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x}^{3} + x \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{14} \cdot 3.936759889140842 \cdot 10^{-8}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.5:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x\_m}^{3} + x\_m \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot 0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.5)
    (/ (+ (* -0.3333333333333333 (pow x_m 3.0)) (* x_m -2.0)) -2.0)
    (* (pow x_m 7.0) 0.0001984126984126984))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.5) {
		tmp = ((-0.3333333333333333 * pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0;
	} else {
		tmp = pow(x_m, 7.0) * 0.0001984126984126984;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5.5d0) then
        tmp = (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + (x_m * (-2.0d0))) / (-2.0d0)
    else
        tmp = (x_m ** 7.0d0) * 0.0001984126984126984d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5.5) {
		tmp = ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0;
	} else {
		tmp = Math.pow(x_m, 7.0) * 0.0001984126984126984;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5.5:
		tmp = ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (x_m * -2.0)) / -2.0
	else:
		tmp = math.pow(x_m, 7.0) * 0.0001984126984126984
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5.5)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * -2.0)) / -2.0);
	else
		tmp = Float64((x_m ^ 7.0) * 0.0001984126984126984);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5.5)
		tmp = ((-0.3333333333333333 * (x_m ^ 3.0)) + (x_m * -2.0)) / -2.0;
	else
		tmp = (x_m ^ 7.0) * 0.0001984126984126984;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5.5], N[(N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.5

   1. Initial program 36.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 36.5%

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

      2. remove-double-neg 36.5%

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

      3. remove-double-neg 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. +-commutative 36.5%

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

      6. sub-neg 36.5%

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

      7. distribute-neg-frac 36.5%

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

      8. distribute-neg-frac2 36.5%

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

      9. remove-double-neg 36.5%

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

      10. metadata-eval 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 92.3%

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

   if 5.5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. +-commutative 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. distribute-neg-frac2 100.0%

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

      9. remove-double-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}}{-2} \]

   6. Taylor expanded in x around inf 95.1%

      \[\leadsto \frac{\color{blue}{-0.0003968253968253968 \cdot {x}^{7}}}{-2} \]
   7. Step-by-step derivation

      1. *-commutative 95.1%

         \[\leadsto \frac{\color{blue}{{x}^{7} \cdot -0.0003968253968253968}}{-2} \]

      2. associate-/l* 95.1%

         \[\leadsto \color{blue}{{x}^{7} \cdot \frac{-0.0003968253968253968}{-2}} \]

      3. metadata-eval 95.1%

         \[\leadsto {x}^{7} \cdot \color{blue}{0.0001984126984126984} \]

   8. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{{x}^{7} \cdot 0.0001984126984126984} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x}^{3} + x \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot 0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2:\\ \;\;\;\;\frac{x\_m \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot 0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4.2)
    (/ (* x_m -2.0) -2.0)
    (* (pow x_m 7.0) 0.0001984126984126984))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4.2) {
		tmp = (x_m * -2.0) / -2.0;
	} else {
		tmp = pow(x_m, 7.0) * 0.0001984126984126984;
	}
	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 <= 4.2d0) then
        tmp = (x_m * (-2.0d0)) / (-2.0d0)
    else
        tmp = (x_m ** 7.0d0) * 0.0001984126984126984d0
    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 <= 4.2) {
		tmp = (x_m * -2.0) / -2.0;
	} else {
		tmp = Math.pow(x_m, 7.0) * 0.0001984126984126984;
	}
	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 <= 4.2:
		tmp = (x_m * -2.0) / -2.0
	else:
		tmp = math.pow(x_m, 7.0) * 0.0001984126984126984
	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 <= 4.2)
		tmp = Float64(Float64(x_m * -2.0) / -2.0);
	else
		tmp = Float64((x_m ^ 7.0) * 0.0001984126984126984);
	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 <= 4.2)
		tmp = (x_m * -2.0) / -2.0;
	else
		tmp = (x_m ^ 7.0) * 0.0001984126984126984;
	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, 4.2], N[(N[(x$95$m * -2.0), $MachinePrecision] / -2.0), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 36.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 36.5%

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

      2. remove-double-neg 36.5%

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

      3. remove-double-neg 36.5%

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

      4. distribute-neg-in 36.5%

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

      5. +-commutative 36.5%

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

      6. sub-neg 36.5%

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

      7. distribute-neg-frac 36.5%

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

      8. distribute-neg-frac2 36.5%

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

      9. remove-double-neg 36.5%

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

      10. metadata-eval 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.1%

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

   if 4.20000000000000018 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. +-commutative 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. distribute-neg-frac2 100.0%

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

      9. remove-double-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}}{-2} \]

   6. Taylor expanded in x around inf 95.1%

      \[\leadsto \frac{\color{blue}{-0.0003968253968253968 \cdot {x}^{7}}}{-2} \]
   7. Step-by-step derivation

      1. *-commutative 95.1%

         \[\leadsto \frac{\color{blue}{{x}^{7} \cdot -0.0003968253968253968}}{-2} \]

      2. associate-/l* 95.1%

         \[\leadsto \color{blue}{{x}^{7} \cdot \frac{-0.0003968253968253968}{-2}} \]

      3. metadata-eval 95.1%

         \[\leadsto {x}^{7} \cdot \color{blue}{0.0001984126984126984} \]

   8. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{{x}^{7} \cdot 0.0001984126984126984} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;\frac{x \cdot -2}{-2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot 0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.2% accurate, 41.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (* x_m -2.0) -2.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * -2.0) / -2.0);
}

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 * (-2.0d0)) / (-2.0d0))
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 * -2.0) / -2.0);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * -2.0) / -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(x_m * -2.0) / -2.0))
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 * -2.0) / -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[(x$95$m * -2.0), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Step-by-step derivation

   1. sub-neg 50.8%

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

   2. remove-double-neg 50.8%

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

   3. remove-double-neg 50.8%

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

   4. distribute-neg-in 50.8%

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

   5. +-commutative 50.8%

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

   6. sub-neg 50.8%

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

   7. distribute-neg-frac 50.8%

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

   8. distribute-neg-frac2 50.8%

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

   9. remove-double-neg 50.8%

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

   10. metadata-eval 50.8%

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

3. Simplified 50.8%

   \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 55.4%

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

6. Final simplification 55.4%

   \[\leadsto \frac{x \cdot -2}{-2} \]
7. Add Preprocessing

Alternative 8: 4.3% accurate, 206.0× 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 50.8%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Step-by-step derivation

   1. sub-neg 50.8%

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

   2. remove-double-neg 50.8%

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

   3. remove-double-neg 50.8%

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

   4. distribute-neg-in 50.8%

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

   5. +-commutative 50.8%

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

   6. sub-neg 50.8%

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

   7. distribute-neg-frac 50.8%

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

   8. distribute-neg-frac2 50.8%

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

   9. remove-double-neg 50.8%

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

   10. metadata-eval 50.8%

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

3. Simplified 50.8%

   \[\leadsto \color{blue}{\frac{e^{-x} - e^{x}}{-2}} \]
4. Add Preprocessing

6. Applied egg-rr 2.8%

   \[\leadsto \frac{\color{blue}{-2}}{-2} \]

7. Final simplification 2.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))