Hyperbolic sine

Percentage Accurate: 55.1% → 100.0%

Time: 8.0s

Alternatives: 8

Speedup: 1.9×

• 50.4% 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: 55.1% 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, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, \mathsf{fma}\left(0.016666666666666666, {x\_m}^{4}, 2\right), \mathsf{fma}\left(0.0003968253968253968, {x\_m}^{4}, 0.3333333333333333\right) \cdot {x\_m}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.1)
      (/
       (fma
        x_m
        (fma 0.016666666666666666 (pow x_m 4.0) 2.0)
        (*
         (fma 0.0003968253968253968 (pow x_m 4.0) 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 <= 0.1) {
		tmp = fma(x_m, fma(0.016666666666666666, pow(x_m, 4.0), 2.0), (fma(0.0003968253968253968, pow(x_m, 4.0), 0.3333333333333333) * 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 <= 0.1)
		tmp = Float64(fma(x_m, fma(0.016666666666666666, (x_m ^ 4.0), 2.0), Float64(fma(0.0003968253968253968, (x_m ^ 4.0), 0.3333333333333333) * (x_m ^ 3.0))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(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[(x$95$m * N[(0.016666666666666666 * N[Power[x$95$m, 4.0], $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.0003968253968253968 * N[Power[x$95$m, 4.0], $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * 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))) < 0.10000000000000001

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0 94.1%

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

      1. distribute-rgt-in 94.1%

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

      2. *-commutative 94.1%

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

      3. associate-*r* 94.1%

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

      4. +-commutative 94.1%

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

      5. distribute-lft-in 94.1%

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

      6. *-commutative 94.1%

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

      7. associate-+l+ 94.1%

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

      8. distribute-lft-in 94.1%

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

      9. associate-*r* 94.1%

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

      10. associate-+r+ 94.1%

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

   5. Simplified 94.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.016666666666666666, {x}^{4}, 2\right), \mathsf{fma}\left(0.0003968253968253968, {x}^{4}, 0.3333333333333333\right) \cdot {x}^{3}\right)}}{2} \]

   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 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.016666666666666666, {x}^{4}, 2\right), \mathsf{fma}\left(0.0003968253968253968, {x}^{4}, 0.3333333333333333\right) \cdot {x}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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
         (*
          (pow x_m 2.0)
          (+
           0.3333333333333333
           (*
            (pow x_m 2.0)
            (+
             0.016666666666666666
             (* 0.0003968253968253968 (pow 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 = (x_m * (2.0 + (pow(x_m, 2.0) * (0.3333333333333333 + (pow(x_m, 2.0) * (0.016666666666666666 + (0.0003968253968253968 * pow(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 = (x_m * (2.0d0 + ((x_m ** 2.0d0) * (0.3333333333333333d0 + ((x_m ** 2.0d0) * (0.016666666666666666d0 + (0.0003968253968253968d0 * (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 = (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.3333333333333333 + (Math.pow(x_m, 2.0) * (0.016666666666666666 + (0.0003968253968253968 * Math.pow(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 = (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.3333333333333333 + (math.pow(x_m, 2.0) * (0.016666666666666666 + (0.0003968253968253968 * math.pow(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(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.3333333333333333 + Float64((x_m ^ 2.0) * Float64(0.016666666666666666 + Float64(0.0003968253968253968 * (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 = (x_m * (2.0 + ((x_m ^ 2.0) * (0.3333333333333333 + ((x_m ^ 2.0) * (0.016666666666666666 + (0.0003968253968253968 * (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[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.016666666666666666 + N[(0.0003968253968253968 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 40.4%

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

   3. Taylor expanded in x around 0 94.1%

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

      1. *-commutative 94.1%

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

   5. Simplified 94.1%

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

   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 95.5%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (* x_s (if (<= t_0 1e-10) (/ (* 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-10) {
		tmp = (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-10) then
        tmp = (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-10) {
		tmp = (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-10:
		tmp = (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-10)
		tmp = Float64(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-10)
		tmp = (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-10], N[(N[(x$95$m * 2.0), $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))) < 1.00000000000000004e-10

   1. Initial program 40.1%

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

   3. Taylor expanded in x around 0 65.4%

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

   if 1.00000000000000004e-10 < (-.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 74.1%

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

Alternative 4: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.1:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x\_m}^{7}}{2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4.1)
    (/ (* x_m 2.0) 2.0)
    (/ (* 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) {
	double tmp;
	if (x_m <= 4.1) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = (0.0003968253968253968 * pow(x_m, 7.0)) / 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 <= 4.1d0) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = (0.0003968253968253968d0 * (x_m ** 7.0d0)) / 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 <= 4.1) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = (0.0003968253968253968 * Math.pow(x_m, 7.0)) / 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 <= 4.1:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = (0.0003968253968253968 * math.pow(x_m, 7.0)) / 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 <= 4.1)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64(Float64(0.0003968253968253968 * (x_m ^ 7.0)) / 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 <= 4.1)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = (0.0003968253968253968 * (x_m ^ 7.0)) / 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, 4.1], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.0999999999999996

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0 65.3%

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

   if 4.0999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 89.6%

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

      1. *-commutative 89.6%

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

   5. Simplified 89.6%

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

   6. Taylor expanded in x around inf 89.6%

      \[\leadsto \frac{\color{blue}{0.0003968253968253968 \cdot {x}^{7}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot \left(2 + 0.0003968253968253968 \cdot {x\_m}^{6}\right)}{2} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (* x_m (+ 2.0 (* 0.0003968253968253968 (pow x_m 6.0)))) 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 + (0.0003968253968253968 * pow(x_m, 6.0)))) / 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 + (0.0003968253968253968d0 * (x_m ** 6.0d0)))) / 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 + (0.0003968253968253968 * Math.pow(x_m, 6.0)))) / 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 + (0.0003968253968253968 * math.pow(x_m, 6.0)))) / 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 * Float64(2.0 + Float64(0.0003968253968253968 * (x_m ^ 6.0)))) / 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 + (0.0003968253968253968 * (x_m ^ 6.0)))) / 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 * N[(2.0 + N[(0.0003968253968253968 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in x around 0 93.0%

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

   1. *-commutative 93.0%

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

5. Simplified 93.0%

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

6. Taylor expanded in x around inf 92.7%

   \[\leadsto \frac{x \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {x}^{6}}\right)}{2} \]

7. Final simplification 92.7%

   \[\leadsto \frac{x \cdot \left(2 + 0.0003968253968253968 \cdot {x}^{6}\right)}{2} \]
8. Add Preprocessing

Alternative 6: 51.4% 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 #s(literal 1 binary64) 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 55.0%

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

3. Taylor expanded in x around 0 50.6%

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

4. Final simplification 50.6%

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

Alternative 7: 3.4% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.0;
}

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 * 0.0d0
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 * 0.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

4. Applied egg-rr 3.2%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

5. Final simplification 3.2%

   \[\leadsto 0 \]
6. Add Preprocessing

Alternative 8: 4.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0.25 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.25))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.25;
}

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 * 0.25d0
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 * 0.25;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.25

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.25)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.25;
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 * 0.25), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

4. Applied egg-rr 2.8%

   \[\leadsto \frac{\color{blue}{0.5}}{2} \]

5. Final simplification 2.8%

   \[\leadsto 0.25 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))