Hyperbolic sine

Percentage Accurate: 54.0% → 100.0%

Time: 7.8s

Alternatives: 9

Speedup: 1.9×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.05)
      (/
       (+
        (* 0.0003968253968253968 (pow x_m 7.0))
        (+
         (* 0.016666666666666666 (pow x_m 5.0))
         (+ (* 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 <= 0.05) {
		tmp = ((0.0003968253968253968 * pow(x_m, 7.0)) + ((0.016666666666666666 * pow(x_m, 5.0)) + ((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 <= 0.05d0) then
        tmp = ((0.0003968253968253968d0 * (x_m ** 7.0d0)) + ((0.016666666666666666d0 * (x_m ** 5.0d0)) + ((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 <= 0.05) {
		tmp = ((0.0003968253968253968 * Math.pow(x_m, 7.0)) + ((0.016666666666666666 * Math.pow(x_m, 5.0)) + ((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 <= 0.05:
		tmp = ((0.0003968253968253968 * math.pow(x_m, 7.0)) + ((0.016666666666666666 * math.pow(x_m, 5.0)) + ((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 <= 0.05)
		tmp = Float64(Float64(Float64(0.0003968253968253968 * (x_m ^ 7.0)) + Float64(Float64(0.016666666666666666 * (x_m ^ 5.0)) + 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 <= 0.05)
		tmp = ((0.0003968253968253968 * (x_m ^ 7.0)) + ((0.016666666666666666 * (x_m ^ 5.0)) + ((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, 0.05], N[(N[(N[(0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $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 42.0%

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

   2. Taylor expanded in x around 0 94.2%

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

   if 0.050000000000000003 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.05:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (((0.0003968253968253968 * Math.pow(x_m, 7.0)) + (Math.log((1.0 + Math.expm1((0.016666666666666666 * Math.pow(x_m, 5.0))))) + ((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 * (((0.0003968253968253968 * math.pow(x_m, 7.0)) + (math.log((1.0 + math.expm1((0.016666666666666666 * math.pow(x_m, 5.0))))) + ((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(0.0003968253968253968 * (x_m ^ 7.0)) + Float64(log(Float64(1.0 + expm1(Float64(0.016666666666666666 * (x_m ^ 5.0))))) + Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + Float64(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[(0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(1.0 + N[(Exp[N[(0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in x around 0 92.3%

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

   1. log1p-expm1-u 99.8%

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

   2. log1p-udef 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x_m} - e^{-x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x_m, 0.016666666666666666 \cdot {x_m}^{5} + 0.3333333333333333 \cdot {x_m}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.001)
      (/
       (fma
        2.0
        x_m
        (+
         (* 0.016666666666666666 (pow x_m 5.0))
         (* 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 <= 0.001) {
		tmp = fma(2.0, x_m, ((0.016666666666666666 * pow(x_m, 5.0)) + (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 <= 0.001)
		tmp = Float64(fma(2.0, x_m, Float64(Float64(0.016666666666666666 * (x_m ^ 5.0)) + Float64(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, 0.001], N[(N[(2.0 * x$95$m + N[(N[(0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 1e-3

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0 93.0%

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

      1. associate-+r+ 93.0%

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

      2. +-commutative 93.0%

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

      3. fma-def 93.0%

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

      4. +-commutative 93.0%

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

      5. fma-def 93.0%

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

   4. Simplified 93.0%

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

      1. fma-udef 93.0%

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

   6. Applied egg-rr 93.0%

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

   if 1e-3 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 0.016666666666666666 \cdot {x}^{5} + 0.3333333333333333 \cdot {x}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]

Alternative 4: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x_m} - e^{-x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 0.001:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x_m}^{3} + x_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.001)
      (/ (+ (* 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 <= 0.001) {
		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 <= 0.001d0) 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 <= 0.001) {
		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 <= 0.001:
		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 <= 0.001)
		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 <= 0.001)
		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, 0.001], 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))) < 1e-3

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0 87.6%

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

   if 1e-3 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.001:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 5.5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x_m}^{3} + x_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x_m}^{7}}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.5)
    (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* 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 <= 5.5) {
		tmp = ((0.3333333333333333 * pow(x_m, 3.0)) + (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 <= 5.5d0) then
        tmp = ((0.3333333333333333d0 * (x_m ** 3.0d0)) + (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 <= 5.5) {
		tmp = ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (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 <= 5.5:
		tmp = ((0.3333333333333333 * math.pow(x_m, 3.0)) + (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 <= 5.5)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + 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 <= 5.5)
		tmp = ((0.3333333333333333 * (x_m ^ 3.0)) + (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, 5.5], N[(N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(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 < 5.5

   1. Initial program 42.3%

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

   2. Taylor expanded in x around 0 87.2%

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

   if 5.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.1%

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

   3. Taylor expanded in x around inf 87.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7}}{2}\\ \end{array} \]

Alternative 6: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 4.2:\\ \;\;\;\;\frac{x_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x_m}^{7}}{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4.2)
    (/ (* 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 <= 4.2) {
		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 <= 4.2d0) 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 <= 4.2) {
		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 <= 4.2:
		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 <= 4.2)
		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 <= 4.2)
		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, 4.2], 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 < 4.20000000000000018

   1. Initial program 42.3%

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

   2. Taylor expanded in x around 0 64.0%

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

   if 4.20000000000000018 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.1%

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

   3. Taylor expanded in x around inf 87.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7}}{2}\\ \end{array} \]

Alternative 7: 52.4% 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 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 55.6%

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

2. Taylor expanded in x around 0 50.6%

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

3. Final simplification 50.6%

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

Alternative 8: 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 55.6%

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

3. Applied egg-rr 3.4%

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

4. Final simplification 3.4%

   \[\leadsto 0 \]

Alternative 9: 4.3% accurate, 206.0× speedup

Math

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

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 55.6%

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

3. Applied egg-rr 2.8%

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

4. Final simplification 2.8%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023331 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))