Hyperbolic sine

Percentage Accurate: 54.3% → 100.0%

Time: 7.2s

Alternatives: 11

Speedup: 29.4×

• 49.7% 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: 54.3% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x\_m, \left(0.3333333333333333 \cdot {x\_m}^{3} + 0.016666666666666666 \cdot {x\_m}^{5}\right) + 0.0003968253968253968 \cdot {x\_m}^{7}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.05)
      (/
       (fma
        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 <= 0.05) {
		tmp = fma(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;
}

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 <= 0.05)
		tmp = Float64(fma(2.0, x_m, Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + 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

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, 0.05], N[(N[(2.0 * x$95$m + N[(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] + N[(0.0003968253968253968 * N[Power[x$95$m, 7.0], $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))) < 0.050000000000000003

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0 95.1%

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

   5. Simplified 95.1%

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

      1. fma-undefine 95.1%

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

      2. fma-undefine 95.1%

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

      3. associate-+r+ 95.1%

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

   7. Applied egg-rr 95.1%

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

   if 0.050000000000000003 < (-.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. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left(0.16666666666666666 + {x\_m}^{2} \cdot \left(0.008333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.05)
      (*
       x_m
       (+
        1.0
        (*
         (pow x_m 2.0)
         (+
          0.16666666666666666
          (*
           (pow x_m 2.0)
           (+ 0.008333333333333333 (* (* x_m x_m) 0.0001984126984126984)))))))
      (/ 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 <= 0.05) {
		tmp = x_m * (1.0 + (pow(x_m, 2.0) * (0.16666666666666666 + (pow(x_m, 2.0) * (0.008333333333333333 + ((x_m * x_m) * 0.0001984126984126984))))));
	} 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 <= 0.05d0) then
        tmp = x_m * (1.0d0 + ((x_m ** 2.0d0) * (0.16666666666666666d0 + ((x_m ** 2.0d0) * (0.008333333333333333d0 + ((x_m * x_m) * 0.0001984126984126984d0))))))
    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 <= 0.05) {
		tmp = x_m * (1.0 + (Math.pow(x_m, 2.0) * (0.16666666666666666 + (Math.pow(x_m, 2.0) * (0.008333333333333333 + ((x_m * x_m) * 0.0001984126984126984))))));
	} 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 <= 0.05:
		tmp = x_m * (1.0 + (math.pow(x_m, 2.0) * (0.16666666666666666 + (math.pow(x_m, 2.0) * (0.008333333333333333 + ((x_m * x_m) * 0.0001984126984126984))))))
	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 <= 0.05)
		tmp = Float64(x_m * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(0.16666666666666666 + Float64((x_m ^ 2.0) * Float64(0.008333333333333333 + Float64(Float64(x_m * x_m) * 0.0001984126984126984)))))));
	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 <= 0.05)
		tmp = x_m * (1.0 + ((x_m ^ 2.0) * (0.16666666666666666 + ((x_m ^ 2.0) * (0.008333333333333333 + ((x_m * x_m) * 0.0001984126984126984))))));
	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, 0.05], N[(x$95$m * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.050000000000000003

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0 95.1%

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

   5. Simplified 95.1%

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

   6. Taylor expanded in x around 0 95.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 + 0.0001984126984126984 \cdot {x}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 95.1%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 + \color{blue}{{x}^{2} \cdot 0.0001984126984126984}\right)\right)\right) \]

   8. Simplified 95.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 + {x}^{2} \cdot 0.0001984126984126984\right)\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 95.1%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0001984126984126984\right)\right)\right) \]

   10. Applied egg-rr 95.1%

       \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0001984126984126984\right)\right)\right) \]

   if 0.050000000000000003 < (-.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. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left(0.16666666666666666 + {x\_m}^{2} \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.02)
      (*
       x_m
       (+
        1.0
        (*
         (pow x_m 2.0)
         (+ 0.16666666666666666 (* (pow x_m 2.0) 0.008333333333333333)))))
      (/ 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 <= 0.02) {
		tmp = x_m * (1.0 + (pow(x_m, 2.0) * (0.16666666666666666 + (pow(x_m, 2.0) * 0.008333333333333333))));
	} 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 <= 0.02d0) then
        tmp = x_m * (1.0d0 + ((x_m ** 2.0d0) * (0.16666666666666666d0 + ((x_m ** 2.0d0) * 0.008333333333333333d0))))
    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 <= 0.02) {
		tmp = x_m * (1.0 + (Math.pow(x_m, 2.0) * (0.16666666666666666 + (Math.pow(x_m, 2.0) * 0.008333333333333333))));
	} 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 <= 0.02:
		tmp = x_m * (1.0 + (math.pow(x_m, 2.0) * (0.16666666666666666 + (math.pow(x_m, 2.0) * 0.008333333333333333))))
	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 <= 0.02)
		tmp = Float64(x_m * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(0.16666666666666666 + Float64((x_m ^ 2.0) * 0.008333333333333333)))));
	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 <= 0.02)
		tmp = x_m * (1.0 + ((x_m ^ 2.0) * (0.16666666666666666 + ((x_m ^ 2.0) * 0.008333333333333333))));
	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, 0.02], N[(x$95$m * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0200000000000000004

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0 95.1%

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

   5. Simplified 95.1%

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

   6. Taylor expanded in x around 0 94.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 94.1%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{{x}^{2} \cdot 0.008333333333333333}\right)\right) \]

   8. Simplified 94.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot 0.008333333333333333\right)\right)} \]

   if 0.0200000000000000004 < (-.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. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m \cdot 2 + x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.3333333333333333\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 2e-6)
      (/ (+ (* x_m 2.0) (* x_m (* x_m (* x_m 0.3333333333333333)))) 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 <= 2e-6) {
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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 <= 2d-6) then
        tmp = ((x_m * 2.0d0) + (x_m * (x_m * (x_m * 0.3333333333333333d0)))) / 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 <= 2e-6) {
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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 <= 2e-6:
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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 <= 2e-6)
		tmp = Float64(Float64(Float64(x_m * 2.0) + Float64(x_m * Float64(x_m * Float64(x_m * 0.3333333333333333)))) / 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 <= 2e-6)
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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, 2e-6], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.3333333333333333), $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))) < 1.99999999999999991e-6

   1. Initial program 33.0%

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

   3. Taylor expanded in x around 0 90.6%

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

      1. *-un-lft-identity 90.6%

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

      2. *-commutative 90.6%

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

      3. metadata-eval 90.6%

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

      4. associate-/l* 90.6%

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

      5. *-commutative 90.6%

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

      6. pow2 90.6%

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

      7. clear-num 90.6%

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

      8. un-div-inv 90.6%

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

      9. *-commutative 90.6%

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

      10. associate-/l* 90.6%

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

      11. metadata-eval 90.6%

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

      12. *-commutative 90.6%

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

      13. *-un-lft-identity 90.6%

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

      14. associate-/r* 90.6%

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

      15. metadata-eval 90.6%

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

   5. Applied egg-rr 90.6%

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

      1. associate-*r/ 90.6%

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

      2. clear-num 90.6%

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

   7. Applied egg-rr 90.6%

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

      1. distribute-lft-in 90.6%

         \[\leadsto \frac{\color{blue}{x \cdot 2 + x \cdot \frac{1}{\frac{\frac{1}{x}}{0.3333333333333333 \cdot x}}}}{2} \]

      2. associate-/r/ 90.6%

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

      3. inv-pow 90.6%

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

      4. pow-flip 90.6%

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

      5. metadata-eval 90.6%

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

      6. pow1 90.6%

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

      7. *-commutative 90.6%

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

   9. Applied egg-rr 90.6%

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

   if 1.99999999999999991e-6 < (-.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. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\frac{x\_m \cdot 2 + x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.3333333333333333\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x\_m} + -1}{2}\\ \end{array} \end{array} \]

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 2.2)
    (/ (+ (* x_m 2.0) (* x_m (* x_m (* x_m 0.3333333333333333)))) 2.0)
    (/ (+ (exp x_m) -1.0) 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 <= 2.2) {
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 2.0;
	} else {
		tmp = (exp(x_m) + -1.0) / 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 <= 2.2d0) then
        tmp = ((x_m * 2.0d0) + (x_m * (x_m * (x_m * 0.3333333333333333d0)))) / 2.0d0
    else
        tmp = (exp(x_m) + (-1.0d0)) / 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 <= 2.2) {
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 2.0;
	} else {
		tmp = (Math.exp(x_m) + -1.0) / 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 <= 2.2:
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 2.0
	else:
		tmp = (math.exp(x_m) + -1.0) / 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 <= 2.2)
		tmp = Float64(Float64(Float64(x_m * 2.0) + Float64(x_m * Float64(x_m * Float64(x_m * 0.3333333333333333)))) / 2.0);
	else
		tmp = Float64(Float64(exp(x_m) + -1.0) / 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 <= 2.2)
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 2.0;
	else
		tmp = (exp(x_m) + -1.0) / 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, 2.2], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 33.6%

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

   3. Taylor expanded in x around 0 90.2%

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

      1. *-un-lft-identity 90.2%

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

      2. *-commutative 90.2%

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

      3. metadata-eval 90.2%

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

      4. associate-/l* 90.2%

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

      5. *-commutative 90.2%

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

      6. pow2 90.2%

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

      7. clear-num 90.2%

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

      8. un-div-inv 90.2%

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

      9. *-commutative 90.2%

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

      10. associate-/l* 90.2%

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

      11. metadata-eval 90.2%

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

      12. *-commutative 90.2%

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

      13. *-un-lft-identity 90.2%

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

      14. associate-/r* 90.2%

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

      15. metadata-eval 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.2%

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

      2. clear-num 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. distribute-lft-in 90.2%

         \[\leadsto \frac{\color{blue}{x \cdot 2 + x \cdot \frac{1}{\frac{\frac{1}{x}}{0.3333333333333333 \cdot x}}}}{2} \]

      2. associate-/r/ 90.2%

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

      3. inv-pow 90.2%

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

      4. pow-flip 90.2%

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

      5. metadata-eval 90.2%

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

      6. pow1 90.2%

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

      7. *-commutative 90.2%

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

   9. Applied egg-rr 90.2%

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

   if 2.2000000000000002 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   5. Simplified 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{1}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{x \cdot 2 + x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.8:\\ \;\;\;\;\frac{x\_m \cdot 2 + x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot 0.3333333333333333\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot 0.0001984126984126984\\ \end{array} \end{array} \]

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 5.8)
    (/ (+ (* x_m 2.0) (* x_m (* x_m (* x_m 0.3333333333333333)))) 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 <= 5.8) {
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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 <= 5.8d0) then
        tmp = ((x_m * 2.0d0) + (x_m * (x_m * (x_m * 0.3333333333333333d0)))) / 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 <= 5.8) {
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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 <= 5.8:
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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 <= 5.8)
		tmp = Float64(Float64(Float64(x_m * 2.0) + Float64(x_m * Float64(x_m * Float64(x_m * 0.3333333333333333)))) / 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 <= 5.8)
		tmp = ((x_m * 2.0) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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, 5.8], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $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 < 5.79999999999999982

   1. Initial program 33.6%

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

   3. Taylor expanded in x around 0 90.2%

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

      1. *-un-lft-identity 90.2%

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

      2. *-commutative 90.2%

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

      3. metadata-eval 90.2%

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

      4. associate-/l* 90.2%

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

      5. *-commutative 90.2%

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

      6. pow2 90.2%

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

      7. clear-num 90.2%

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

      8. un-div-inv 90.2%

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

      9. *-commutative 90.2%

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

      10. associate-/l* 90.2%

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

      11. metadata-eval 90.2%

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

      12. *-commutative 90.2%

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

      13. *-un-lft-identity 90.2%

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

      14. associate-/r* 90.2%

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

      15. metadata-eval 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.2%

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

      2. clear-num 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. distribute-lft-in 90.2%

         \[\leadsto \frac{\color{blue}{x \cdot 2 + x \cdot \frac{1}{\frac{\frac{1}{x}}{0.3333333333333333 \cdot x}}}}{2} \]

      2. associate-/r/ 90.2%

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

      3. inv-pow 90.2%

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

      4. pow-flip 90.2%

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

      5. metadata-eval 90.2%

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

      6. pow1 90.2%

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

      7. *-commutative 90.2%

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

   9. Applied egg-rr 90.2%

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

   if 5.79999999999999982 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 84.1%

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

   5. Simplified 84.1%

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

   6. Taylor expanded in x around inf 84.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8:\\ \;\;\;\;\frac{x \cdot 2 + x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot 0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 13.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(1 + x\_m \cdot \left(0.25 + x\_m \cdot \left(0.08333333333333333 + x\_m \cdot 0.020833333333333332\right)\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   x_m
   (+
    1.0
    (*
     x_m
     (+ 0.25 (* x_m (+ 0.08333333333333333 (* x_m 0.020833333333333332)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * (1.0d0 + (x_m * (0.25d0 + (x_m * (0.08333333333333333d0 + (x_m * 0.020833333333333332d0)))))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(1.0 + Float64(x_m * Float64(0.25 + Float64(x_m * Float64(0.08333333333333333 + Float64(x_m * 0.020833333333333332))))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * N[(1.0 + N[(x$95$m * N[(0.25 + N[(x$95$m * N[(0.08333333333333333 + N[(x$95$m * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in x around 0 28.6%

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

   1. mul-1-neg 28.6%

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

   2. unsub-neg 28.6%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

5. Simplified 28.6%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

6. Taylor expanded in x around 0 72.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + 0.020833333333333332 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 72.7%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + \color{blue}{x \cdot 0.020833333333333332}\right)\right)\right) \]

8. Simplified 72.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + x \cdot 0.020833333333333332\right)\right)\right)} \]
9. Add Preprocessing

Alternative 8: 84.5% accurate, 15.8× speedup

Math

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

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) (* x_m (* x_m (* x_m 0.3333333333333333)))) 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) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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) + (x_m * (x_m * (x_m * 0.3333333333333333d0)))) / 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) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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(x_m * 2.0) + Float64(x_m * Float64(x_m * Float64(x_m * 0.3333333333333333)))) / 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) + (x_m * (x_m * (x_m * 0.3333333333333333)))) / 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[(x$95$m * 2.0), $MachinePrecision] + N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in x around 0 84.7%

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

   1. *-un-lft-identity 84.7%

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

   2. *-commutative 84.7%

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

   3. metadata-eval 84.7%

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

   4. associate-/l* 84.7%

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

   5. *-commutative 84.7%

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

   6. pow2 84.7%

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

   7. clear-num 84.7%

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

   8. un-div-inv 84.7%

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

   9. *-commutative 84.7%

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

   10. associate-/l* 84.7%

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

   11. metadata-eval 84.7%

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

   12. *-commutative 84.7%

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

   13. *-un-lft-identity 84.7%

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

   14. associate-/r* 84.7%

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

   15. metadata-eval 84.7%

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

5. Applied egg-rr 84.7%

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

   1. associate-*r/ 84.7%

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

   2. clear-num 84.7%

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

7. Applied egg-rr 84.7%

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

   1. distribute-lft-in 84.7%

      \[\leadsto \frac{\color{blue}{x \cdot 2 + x \cdot \frac{1}{\frac{\frac{1}{x}}{0.3333333333333333 \cdot x}}}}{2} \]

   2. associate-/r/ 84.7%

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

   3. inv-pow 84.7%

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

   4. pow-flip 84.7%

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

   5. metadata-eval 84.7%

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

   6. pow1 84.7%

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

   7. *-commutative 84.7%

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

9. Applied egg-rr 84.7%

   \[\leadsto \frac{\color{blue}{x \cdot 2 + x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]
10. Add Preprocessing

Alternative 9: 84.5% accurate, 18.7× speedup

Math

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

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 (* 0.3333333333333333 (* x_m 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 * ((x_m * (2.0 + (0.3333333333333333 * (x_m * 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 * ((x_m * (2.0d0 + (0.3333333333333333d0 * (x_m * 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 * ((x_m * (2.0 + (0.3333333333333333 * (x_m * 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 * ((x_m * (2.0 + (0.3333333333333333 * (x_m * 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(x_m * Float64(2.0 + Float64(0.3333333333333333 * Float64(x_m * 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 * ((x_m * (2.0 + (0.3333333333333333 * (x_m * 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[(x$95$m * N[(2.0 + N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in x around 0 84.7%

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

   1. unpow2 92.4%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0001984126984126984\right)\right)\right) \]

5. Applied egg-rr 84.7%

   \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{2} \]
6. Add Preprocessing

Alternative 10: 75.7% accurate, 29.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m (+ 1.0 (* x_m 0.25)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * 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 * (x_m * (1.0d0 + (x_m * 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 * (x_m * (1.0 + (x_m * 0.25)));
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(1.0 + Float64(x_m * 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 * (x_m * (1.0 + (x_m * 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 * N[(x$95$m * N[(1.0 + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in x around 0 28.6%

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

   1. mul-1-neg 28.6%

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

   2. unsub-neg 28.6%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

5. Simplified 28.6%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

6. Taylor expanded in x around 0 68.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.25 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-commutative 68.3%

      \[\leadsto x \cdot \left(1 + \color{blue}{x \cdot 0.25}\right) \]

8. Simplified 68.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot 0.25\right)} \]
9. Add Preprocessing

Alternative 11: 52.2% accurate, 206.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in x around 0 92.4%

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

5. Simplified 92.4%

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

6. Taylor expanded in x around 0 57.3%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))