Hyperbolic sine

Percentage Accurate: 54.9% → 100.0%

Time: 3.3s

Alternatives: 7

Speedup: 1.9×

• 50.0% 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: 54.9% 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, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x_m} - e^{-x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x_m}^{3} + x_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.0005)
      (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* x_m 2.0)) 2.0)
      (/ t_0 2.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = exp(x_m) - exp(-x_m);
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = ((0.3333333333333333 * pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x_m) - exp(-x_m)
    if (t_0 <= 0.0005d0) then
        tmp = ((0.3333333333333333d0 * (x_m ** 3.0d0)) + (x_m * 2.0d0)) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.exp(x_m) - Math.exp(-x_m);
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.exp(x_m) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 0.0005:
		tmp = ((0.3333333333333333 * math.pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 0.0005)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * 2.0)) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = exp(x_m) - exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 0.0005)
		tmp = ((0.3333333333333333 * (x_m ^ 3.0)) + (x_m * 2.0)) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0005], N[(N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 5.0000000000000001e-4

   1. Initial program 42.1%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 90.6%

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

   if 5.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x_m}^{3} + x_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x_m}^{9} \cdot 0.004629629629629629}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e+22)
    (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* x_m 2.0)) 2.0)
    (cbrt (* (pow x_m 9.0) 0.004629629629629629)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5e+22) {
		tmp = ((0.3333333333333333 * pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0;
	} else {
		tmp = cbrt((pow(x_m, 9.0) * 0.004629629629629629));
	}
	return x_s * tmp;
}

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 <= 5e+22) {
		tmp = ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0;
	} else {
		tmp = Math.cbrt((Math.pow(x_m, 9.0) * 0.004629629629629629));
	}
	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 <= 5e+22)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * 2.0)) / 2.0);
	else
		tmp = cbrt(Float64((x_m ^ 9.0) * 0.004629629629629629));
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 5e+22], N[(N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Power[x$95$m, 9.0], $MachinePrecision] * 0.004629629629629629), $MachinePrecision], 1/3], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999996e22

   1. Initial program 43.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 88.9%

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

   if 4.9999999999999996e22 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 74.2%

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

   3. Taylor expanded in x around inf 74.2%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3}}}{2} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.3333333333333333 \cdot {x}^{3}}{2} \cdot \frac{0.3333333333333333 \cdot {x}^{3}}{2}\right) \cdot \frac{0.3333333333333333 \cdot {x}^{3}}{2}}} \]

      2. pow1/3 100.0%

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

      3. pow3 100.0%

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

      4. div-inv 100.0%

         \[\leadsto {\left({\color{blue}{\left(\left(0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 100.0%

         \[\leadsto {\left({\left(\color{blue}{\left({x}^{3} \cdot 0.3333333333333333\right)} \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-*l* 100.0%

         \[\leadsto {\left({\color{blue}{\left({x}^{3} \cdot \left(0.3333333333333333 \cdot \frac{1}{2}\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      7. unpow-prod-down 100.0%

         \[\leadsto {\color{blue}{\left({\left({x}^{3}\right)}^{3} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      8. pow-pow 100.0%

         \[\leadsto {\left(\color{blue}{{x}^{\left(3 \cdot 3\right)}} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      9. metadata-eval 100.0%

         \[\leadsto {\left({x}^{\color{blue}{9}} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 100.0%

          \[\leadsto {\left({x}^{9} \cdot {\left(0.3333333333333333 \cdot \color{blue}{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

      11. metadata-eval 100.0%

          \[\leadsto {\left({x}^{9} \cdot {\color{blue}{0.16666666666666666}}^{3}\right)}^{0.3333333333333333} \]

      12. metadata-eval 100.0%

          \[\leadsto {\left({x}^{9} \cdot \color{blue}{0.004629629629629629}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({x}^{9} \cdot 0.004629629629629629\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}\\ \end{array} \]

Alternative 3: 84.1% accurate, 1.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* 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 * (((0.3333333333333333 * pow(x_m, 3.0)) + (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 * (((0.3333333333333333d0 * (x_m ** 3.0d0)) + (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 * (((0.3333333333333333 * Math.pow(x_m, 3.0)) + (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 * (((0.3333333333333333 * math.pow(x_m, 3.0)) + (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(Float64(0.3333333333333333 * (x_m ^ 3.0)) + 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 * (((0.3333333333333333 * (x_m ^ 3.0)) + (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[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0 85.9%

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

3. Final simplification 85.9%

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

Alternative 4: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 2.5:\\ \;\;\;\;\frac{x_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{3} \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.5) (/ (* x_m 2.0) 2.0) (* (pow x_m 3.0) 0.16666666666666666))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.5) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = pow(x_m, 3.0) * 0.16666666666666666;
	}
	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 <= 2.5d0) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = (x_m ** 3.0d0) * 0.16666666666666666d0
    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 <= 2.5) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = Math.pow(x_m, 3.0) * 0.16666666666666666;
	}
	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 <= 2.5:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = math.pow(x_m, 3.0) * 0.16666666666666666
	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 <= 2.5)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64((x_m ^ 3.0) * 0.16666666666666666);
	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 <= 2.5)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = (x_m ^ 3.0) * 0.16666666666666666;
	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, 2.5], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 42.1%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 64.8%

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

   if 2.5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 69.0%

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

   3. Taylor expanded in x around inf 69.0%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3}}}{2} \]
   4. Step-by-step derivation

      1. div-inv 69.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2}} \]

      2. *-commutative 69.0%

         \[\leadsto \color{blue}{\left({x}^{3} \cdot 0.3333333333333333\right)} \cdot \frac{1}{2} \]

      3. associate-*l* 69.0%

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(0.3333333333333333 \cdot \frac{1}{2}\right)} \]

      4. metadata-eval 69.0%

         \[\leadsto {x}^{3} \cdot \left(0.3333333333333333 \cdot \color{blue}{0.5}\right) \]

      5. metadata-eval 69.0%

         \[\leadsto {x}^{3} \cdot \color{blue}{0.16666666666666666} \]

   5. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{{x}^{3} \cdot 0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \end{array} \]

Alternative 5: 51.8% accurate, 41.2× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((x_m * 2.0d0) / 2.0d0)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((x_m * 2.0) / 2.0);
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m * 2.0) / 2.0);
end

Wolfram

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

Derivation

1. Initial program 54.6%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0 52.0%

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

3. Final simplification 52.0%

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

Alternative 6: 3.5% 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 1 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 54.6%

   \[\frac{e^{x} - e^{-x}}{2} \]

3. Applied egg-rr 3.5%

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

4. Final simplification 3.5%

   \[\leadsto 0 \]

Alternative 7: 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 1 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 54.6%

   \[\frac{e^{x} - e^{-x}}{2} \]

3. Applied egg-rr 2.7%

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

4. Final simplification 2.7%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023338 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))