Hyperbolic sine

Percentage Accurate: 53.7% → 100.0%

Time: 6.2s

Alternatives: 9

Speedup: 1.9×

• 49.4% 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: 53.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{-2 \cdot x\_m + \left(-0.3333333333333333 \cdot {x\_m}^{3} + \left(-0.016666666666666666 \cdot {x\_m}^{5} + -0.0003968253968253968 \cdot {x\_m}^{7}\right)\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.2)
      (/
       (+
        (* -2.0 x_m)
        (+
         (* -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 <= 0.2) {
		tmp = ((-2.0 * x_m) + ((-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 <= 0.2d0) then
        tmp = (((-2.0d0) * x_m) + (((-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 <= 0.2) {
		tmp = ((-2.0 * x_m) + ((-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 <= 0.2:
		tmp = ((-2.0 * x_m) + ((-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 <= 0.2)
		tmp = Float64(Float64(Float64(-2.0 * x_m) + 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 <= 0.2)
		tmp = ((-2.0 * x_m) + ((-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, 0.2], N[(N[(N[(-2.0 * x$95$m), $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))) < 0.20000000000000001

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. remove-double-neg 43.3%

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

      3. remove-double-neg 43.3%

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

      4. distribute-neg-in 43.3%

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

      5. +-commutative 43.3%

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

      6. sub-neg 43.3%

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

      7. distribute-neg-frac 43.3%

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

      8. distribute-neg-frac2 43.3%

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

      9. remove-double-neg 43.3%

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

      10. metadata-eval 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in x around 0 93.5%

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

   if 0.20000000000000001 < (-.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 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.2:\\ \;\;\;\;\frac{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.016666666666666666 \cdot {x}^{5} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{-2 \cdot x\_m + \left(-0.3333333333333333 \cdot {x\_m}^{3} + \left(\log \left(1 + \mathsf{expm1}\left(-0.016666666666666666 \cdot {x\_m}^{5}\right)\right) + -0.0003968253968253968 \cdot {x\_m}^{7}\right)\right)}{-2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+
    (* -2.0 x_m)
    (+
     (* -0.3333333333333333 (pow x_m 3.0))
     (+
      (log (+ 1.0 (expm1 (* -0.016666666666666666 (pow x_m 5.0)))))
      (* -0.0003968253968253968 (pow x_m 7.0)))))
   -2.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((-2.0 * x_m) + ((-0.3333333333333333 * pow(x_m, 3.0)) + (log((1.0 + expm1((-0.016666666666666666 * pow(x_m, 5.0))))) + (-0.0003968253968253968 * pow(x_m, 7.0))))) / -2.0);
}

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 * (((-2.0 * x_m) + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (Math.log((1.0 + Math.expm1((-0.016666666666666666 * Math.pow(x_m, 5.0))))) + (-0.0003968253968253968 * Math.pow(x_m, 7.0))))) / -2.0);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((-2.0 * x_m) + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (math.log((1.0 + math.expm1((-0.016666666666666666 * math.pow(x_m, 5.0))))) + (-0.0003968253968253968 * math.pow(x_m, 7.0))))) / -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(Float64(-2.0 * x_m) + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(log(Float64(1.0 + expm1(Float64(-0.016666666666666666 * (x_m ^ 5.0))))) + Float64(-0.0003968253968253968 * (x_m ^ 7.0))))) / -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[(N[(-2.0 * x$95$m), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(1.0 + N[(Exp[N[(-0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

   1. sub-neg 57.9%

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

   2. remove-double-neg 57.9%

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

   3. remove-double-neg 57.9%

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

   4. distribute-neg-in 57.9%

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

   5. +-commutative 57.9%

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

   6. sub-neg 57.9%

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

   7. distribute-neg-frac 57.9%

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

   8. distribute-neg-frac2 57.9%

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

   9. remove-double-neg 57.9%

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

   10. metadata-eval 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in x around 0 92.6%

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

   1. log1p-expm1-u 99.6%

      \[\leadsto \frac{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.016666666666666666 \cdot {x}^{5}\right)\right)} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}{-2} \]

   2. log1p-undefine 99.6%

      \[\leadsto \frac{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.016666666666666666 \cdot {x}^{5}\right)\right)} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}{-2} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.016666666666666666 \cdot {x}^{5}\right)\right)} + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}{-2} \]

8. Final simplification 99.6%

   \[\leadsto \frac{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(\log \left(1 + \mathsf{expm1}\left(-0.016666666666666666 \cdot {x}^{5}\right)\right) + -0.0003968253968253968 \cdot {x}^{7}\right)\right)}{-2} \]
9. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{-2 \cdot x\_m + \left(-0.3333333333333333 \cdot {x\_m}^{3} + -0.016666666666666666 \cdot {x\_m}^{5}\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.01)
      (/
       (+
        (* -2.0 x_m)
        (+
         (* -0.3333333333333333 (pow x_m 3.0))
         (* -0.016666666666666666 (pow x_m 5.0))))
       -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 <= 0.01) {
		tmp = ((-2.0 * x_m) + ((-0.3333333333333333 * pow(x_m, 3.0)) + (-0.016666666666666666 * pow(x_m, 5.0)))) / -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 <= 0.01d0) then
        tmp = (((-2.0d0) * x_m) + (((-0.3333333333333333d0) * (x_m ** 3.0d0)) + ((-0.016666666666666666d0) * (x_m ** 5.0d0)))) / (-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 <= 0.01) {
		tmp = ((-2.0 * x_m) + ((-0.3333333333333333 * Math.pow(x_m, 3.0)) + (-0.016666666666666666 * Math.pow(x_m, 5.0)))) / -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 <= 0.01:
		tmp = ((-2.0 * x_m) + ((-0.3333333333333333 * math.pow(x_m, 3.0)) + (-0.016666666666666666 * math.pow(x_m, 5.0)))) / -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 <= 0.01)
		tmp = Float64(Float64(Float64(-2.0 * x_m) + Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) + Float64(-0.016666666666666666 * (x_m ^ 5.0)))) / -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 <= 0.01)
		tmp = ((-2.0 * x_m) + ((-0.3333333333333333 * (x_m ^ 3.0)) + (-0.016666666666666666 * (x_m ^ 5.0)))) / -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, 0.01], N[(N[(N[(-2.0 * x$95$m), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[x$95$m, 5.0], $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))) < 0.0100000000000000002

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. remove-double-neg 43.0%

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

      3. remove-double-neg 43.0%

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

      4. distribute-neg-in 43.0%

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

      5. +-commutative 43.0%

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

      6. sub-neg 43.0%

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

      7. distribute-neg-frac 43.0%

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

      8. distribute-neg-frac2 43.0%

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

      9. remove-double-neg 43.0%

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

      10. metadata-eval 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in x around 0 92.0%

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

   if 0.0100000000000000002 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.01:\\ \;\;\;\;\frac{-2 \cdot x + \left(-0.3333333333333333 \cdot {x}^{3} + -0.016666666666666666 \cdot {x}^{5}\right)}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-2 \cdot x\_m + -0.3333333333333333 \cdot {x\_m}^{3}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 5e-7)
      (/ (+ (* -2.0 x_m) (* -0.3333333333333333 (pow x_m 3.0))) -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 <= 5e-7) {
		tmp = ((-2.0 * x_m) + (-0.3333333333333333 * pow(x_m, 3.0))) / -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 <= 5d-7) then
        tmp = (((-2.0d0) * x_m) + ((-0.3333333333333333d0) * (x_m ** 3.0d0))) / (-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 <= 5e-7) {
		tmp = ((-2.0 * x_m) + (-0.3333333333333333 * Math.pow(x_m, 3.0))) / -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 <= 5e-7:
		tmp = ((-2.0 * x_m) + (-0.3333333333333333 * math.pow(x_m, 3.0))) / -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 <= 5e-7)
		tmp = Float64(Float64(Float64(-2.0 * x_m) + Float64(-0.3333333333333333 * (x_m ^ 3.0))) / -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 <= 5e-7)
		tmp = ((-2.0 * x_m) + (-0.3333333333333333 * (x_m ^ 3.0))) / -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, 5e-7], N[(N[(N[(-2.0 * x$95$m), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $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))) < 4.99999999999999977e-7

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. remove-double-neg 43.0%

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

      3. remove-double-neg 43.0%

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

      4. distribute-neg-in 43.0%

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

      5. +-commutative 43.0%

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

      6. sub-neg 43.0%

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

      7. distribute-neg-frac 43.0%

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

      8. distribute-neg-frac2 43.0%

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

      9. remove-double-neg 43.0%

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

      10. metadata-eval 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in x around 0 84.4%

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

   if 4.99999999999999977e-7 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 88.4%

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

Alternative 5: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.45:\\ \;\;\;\;\frac{-2 \cdot x\_m}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x\_m}^{3}}{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.45)
    (/ (* -2.0 x_m) -2.0)
    (/ (* -0.3333333333333333 (pow x_m 3.0)) -2.0))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.45) {
		tmp = (-2.0 * x_m) / -2.0;
	} else {
		tmp = (-0.3333333333333333 * pow(x_m, 3.0)) / -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) :: tmp
    if (x_m <= 2.45d0) then
        tmp = ((-2.0d0) * x_m) / (-2.0d0)
    else
        tmp = ((-0.3333333333333333d0) * (x_m ** 3.0d0)) / (-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 tmp;
	if (x_m <= 2.45) {
		tmp = (-2.0 * x_m) / -2.0;
	} else {
		tmp = (-0.3333333333333333 * Math.pow(x_m, 3.0)) / -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):
	tmp = 0
	if x_m <= 2.45:
		tmp = (-2.0 * x_m) / -2.0
	else:
		tmp = (-0.3333333333333333 * math.pow(x_m, 3.0)) / -2.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.45)
		tmp = Float64(Float64(-2.0 * x_m) / -2.0);
	else
		tmp = Float64(Float64(-0.3333333333333333 * (x_m ^ 3.0)) / -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)
	tmp = 0.0;
	if (x_m <= 2.45)
		tmp = (-2.0 * x_m) / -2.0;
	else
		tmp = (-0.3333333333333333 * (x_m ^ 3.0)) / -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_] := N[(x$95$s * If[LessEqual[x$95$m, 2.45], N[(N[(-2.0 * x$95$m), $MachinePrecision] / -2.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. remove-double-neg 43.3%

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

      3. remove-double-neg 43.3%

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

      4. distribute-neg-in 43.3%

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

      5. +-commutative 43.3%

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

      6. sub-neg 43.3%

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

      7. distribute-neg-frac 43.3%

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

      8. distribute-neg-frac2 43.3%

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

      9. remove-double-neg 43.3%

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

      10. metadata-eval 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in x around 0 62.9%

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

   if 2.4500000000000002 < 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 68.6%

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

   6. Taylor expanded in x around inf 68.6%

      \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot {x}^{3}}}{-2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;\frac{-2 \cdot x}{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot {x}^{3}}{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((-2.0 * x_m) + (-0.3333333333333333 * pow(x_m, 3.0))) / -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 * ((((-2.0d0) * x_m) + ((-0.3333333333333333d0) * (x_m ** 3.0d0))) / (-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 * (((-2.0 * x_m) + (-0.3333333333333333 * Math.pow(x_m, 3.0))) / -2.0);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((-2.0 * x_m) + (-0.3333333333333333 * math.pow(x_m, 3.0))) / -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(Float64(-2.0 * x_m) + Float64(-0.3333333333333333 * (x_m ^ 3.0))) / -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 * (((-2.0 * x_m) + (-0.3333333333333333 * (x_m ^ 3.0))) / -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[(N[(-2.0 * x$95$m), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

   1. sub-neg 57.9%

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

   2. remove-double-neg 57.9%

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

   3. remove-double-neg 57.9%

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

   4. distribute-neg-in 57.9%

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

   5. +-commutative 57.9%

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

   6. sub-neg 57.9%

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

   7. distribute-neg-frac 57.9%

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

   8. distribute-neg-frac2 57.9%

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

   9. remove-double-neg 57.9%

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

   10. metadata-eval 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in x around 0 80.2%

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

6. Final simplification 80.2%

   \[\leadsto \frac{-2 \cdot x + -0.3333333333333333 \cdot {x}^{3}}{-2} \]
7. Add Preprocessing

Alternative 7: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{-2 \cdot x\_m + -0.016666666666666666 \cdot {x\_m}^{5}}{-2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (+ (* -2.0 x_m) (* -0.016666666666666666 (pow x_m 5.0))) -2.0)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((-2.0 * x_m) + (-0.016666666666666666 * pow(x_m, 5.0))) / -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 * ((((-2.0d0) * x_m) + ((-0.016666666666666666d0) * (x_m ** 5.0d0))) / (-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 * (((-2.0 * x_m) + (-0.016666666666666666 * Math.pow(x_m, 5.0))) / -2.0);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((-2.0 * x_m) + (-0.016666666666666666 * math.pow(x_m, 5.0))) / -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(Float64(-2.0 * x_m) + Float64(-0.016666666666666666 * (x_m ^ 5.0))) / -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 * (((-2.0 * x_m) + (-0.016666666666666666 * (x_m ^ 5.0))) / -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[(N[(-2.0 * x$95$m), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

   1. sub-neg 57.9%

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

   2. remove-double-neg 57.9%

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

   3. remove-double-neg 57.9%

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

   4. distribute-neg-in 57.9%

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

   5. +-commutative 57.9%

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

   6. sub-neg 57.9%

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

   7. distribute-neg-frac 57.9%

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

   8. distribute-neg-frac2 57.9%

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

   9. remove-double-neg 57.9%

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

   10. metadata-eval 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in x around 0 90.3%

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

6. Taylor expanded in x around inf 89.9%

   \[\leadsto \frac{-2 \cdot x + \color{blue}{-0.016666666666666666 \cdot {x}^{5}}}{-2} \]

7. Final simplification 89.9%

   \[\leadsto \frac{-2 \cdot x + -0.016666666666666666 \cdot {x}^{5}}{-2} \]
8. Add Preprocessing

Alternative 8: 52.7% accurate, 41.2× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((-2.0 * x_m) / -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 * (((-2.0d0) * x_m) / (-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 * ((-2.0 * x_m) / -2.0);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((-2.0 * x_m) / -2.0)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(-2.0 * x_m) / -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 * ((-2.0 * x_m) / -2.0);
end

Wolfram

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

Derivation

1. Initial program 57.9%

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

   1. sub-neg 57.9%

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

   2. remove-double-neg 57.9%

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

   3. remove-double-neg 57.9%

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

   4. distribute-neg-in 57.9%

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

   5. +-commutative 57.9%

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

   6. sub-neg 57.9%

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

   7. distribute-neg-frac 57.9%

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

   8. distribute-neg-frac2 57.9%

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

   9. remove-double-neg 57.9%

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

   10. metadata-eval 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in x around 0 48.0%

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

6. Final simplification 48.0%

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

Alternative 9: 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 57.9%

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

   1. sub-neg 57.9%

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

   2. remove-double-neg 57.9%

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

   3. remove-double-neg 57.9%

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

   4. distribute-neg-in 57.9%

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

   5. +-commutative 57.9%

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

   6. sub-neg 57.9%

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

   7. distribute-neg-frac 57.9%

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

   8. distribute-neg-frac2 57.9%

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

   9. remove-double-neg 57.9%

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

   10. metadata-eval 57.9%

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

3. Simplified 57.9%

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

6. Applied egg-rr 2.7%

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

7. Final simplification 2.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))