Hyperbolic sine

Percentage Accurate: 54.0% → 100.0%

Time: 5.4s

Alternatives: 9

Speedup: 1.9×

• 49.3% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, 0.016666666666666666 \cdot {x\_m}^{4} + 2, 0.3333333333333333 \cdot {x\_m}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 0.01)
      (/
       (fma
        x_m
        (+ (* 0.016666666666666666 (pow x_m 4.0)) 2.0)
        (* 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 <= 0.01) {
		tmp = fma(x_m, ((0.016666666666666666 * pow(x_m, 4.0)) + 2.0), (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 <= 0.01)
		tmp = Float64(fma(x_m, Float64(Float64(0.016666666666666666 * (x_m ^ 4.0)) + 2.0), 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, 0.01], N[(N[(x$95$m * N[(N[(0.016666666666666666 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. distribute-lft-in 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-+l+ 92.6%

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

      5. +-commutative 92.6%

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

      6. distribute-lft-in 92.6%

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

      7. *-commutative 92.6%

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

      8. fma-define 92.6%

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

      9. *-commutative 92.6%

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

      10. associate-*l* 92.6%

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

      11. fma-define 92.6%

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

      12. pow-sqr 92.6%

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

      13. metadata-eval 92.6%

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

      14. associate-*l* 92.6%

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

      15. pow-plus 92.6%

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.016666666666666666, {x}^{4}, 2\right), 0.3333333333333333 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)}{2} \]

      16. metadata-eval 92.6%

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

   5. Simplified 92.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.016666666666666666, {x}^{4}, 2\right), 0.3333333333333333 \cdot {x}^{3}\right)}}{2} \]
   6. Step-by-step derivation

      1. fma-undefine 92.6%

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

   7. Applied egg-rr 92.6%

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

   if 0.0100000000000000002 < (-.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

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 0.01)
      (/
       (*
        x_m
        (+
         2.0
         (*
          (pow x_m 2.0)
          (+ 0.3333333333333333 (* 0.016666666666666666 (pow x_m 2.0))))))
       2.0)
      (/ t_0 2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = exp(x_m) - exp(-x_m);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = (x_m * (2.0 + (pow(x_m, 2.0) * (0.3333333333333333 + (0.016666666666666666 * pow(x_m, 2.0)))))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x_m) - exp(-x_m)
    if (t_0 <= 0.01d0) then
        tmp = (x_m * (2.0d0 + ((x_m ** 2.0d0) * (0.3333333333333333d0 + (0.016666666666666666d0 * (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 <= 0.01) {
		tmp = (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.3333333333333333 + (0.016666666666666666 * Math.pow(x_m, 2.0)))))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.exp(x_m) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 0.01:
		tmp = (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.3333333333333333 + (0.016666666666666666 * math.pow(x_m, 2.0)))))) / 2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.3333333333333333 + Float64(0.016666666666666666 * (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 <= 0.01)
		tmp = (x_m * (2.0 + ((x_m ^ 2.0) * (0.3333333333333333 + (0.016666666666666666 * (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, 0.01], N[(N[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(0.016666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 0 92.6%

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

      1. *-commutative 92.6%

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

   5. Simplified 92.6%

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

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

   1. Initial program 100.0%

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

4. Final simplification 94.6%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (* x_s (if (<= t_0 0.0) (/ (* 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 <= 0.0) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x_m) - exp(-x_m)
    if (t_0 <= 0.0d0) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.exp(x_m) - Math.exp(-x_m);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.exp(x_m) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = exp(x_m) - exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 0 66.5%

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

   if 0.0 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.4:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x\_m}^{5}}{2}\\ \end{array} \end{array} \]

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 3.4)
    (/ (* x_m 2.0) 2.0)
    (/ (* 0.016666666666666666 (pow x_m 5.0)) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 3.4) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = (0.016666666666666666 * pow(x_m, 5.0)) / 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 <= 3.4d0) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = (0.016666666666666666d0 * (x_m ** 5.0d0)) / 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 <= 3.4) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = (0.016666666666666666 * Math.pow(x_m, 5.0)) / 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 <= 3.4:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = (0.016666666666666666 * math.pow(x_m, 5.0)) / 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 <= 3.4)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64(Float64(0.016666666666666666 * (x_m ^ 5.0)) / 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 <= 3.4)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = (0.016666666666666666 * (x_m ^ 5.0)) / 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, 3.4], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 0 66.5%

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

   if 3.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. distribute-lft-in 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-+l+ 81.0%

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

      5. +-commutative 81.0%

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

      6. distribute-lft-in 81.0%

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

      7. *-commutative 81.0%

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

      8. fma-define 81.0%

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

      9. *-commutative 81.0%

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

      10. associate-*l* 81.0%

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

      11. fma-define 81.0%

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

      12. pow-sqr 81.0%

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

      13. metadata-eval 81.0%

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

      14. associate-*l* 81.0%

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

      15. pow-plus 81.0%

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.016666666666666666, {x}^{4}, 2\right), 0.3333333333333333 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)}{2} \]

      16. metadata-eval 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in x around inf 81.0%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot \left(0.016666666666666666 \cdot {x\_m}^{4} + 2\right)}{2} \end{array} \]

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 (+ (* 0.016666666666666666 (pow x_m 4.0)) 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 * ((0.016666666666666666 * pow(x_m, 4.0)) + 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 * ((0.016666666666666666d0 * (x_m ** 4.0d0)) + 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 * ((0.016666666666666666 * Math.pow(x_m, 4.0)) + 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 * ((0.016666666666666666 * math.pow(x_m, 4.0)) + 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 * Float64(Float64(0.016666666666666666 * (x_m ^ 4.0)) + 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 * ((0.016666666666666666 * (x_m ^ 4.0)) + 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 * N[(N[(0.016666666666666666 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in x around 0 89.5%

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

   1. *-commutative 89.5%

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf 89.2%

   \[\leadsto \frac{x \cdot \left(2 + \color{blue}{0.016666666666666666 \cdot {x}^{4}}\right)}{2} \]

7. Final simplification 89.2%

   \[\leadsto \frac{x \cdot \left(0.016666666666666666 \cdot {x}^{4} + 2\right)}{2} \]
8. Add Preprocessing

Alternative 6: 52.7% accurate, 41.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (* x_m 2.0) 2.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in x around 0 50.1%

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

4. Final simplification 50.1%

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

Alternative 7: 4.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 4 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 4.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 4.0;
}

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 * 4.0d0
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 * 4.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 4.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 4.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 4.0;
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 * 4.0), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

4. Applied egg-rr 2.7%

   \[\leadsto \frac{\color{blue}{8}}{2} \]

5. Final simplification 2.7%

   \[\leadsto 4 \]
6. Add Preprocessing

Alternative 8: 4.3% accurate, 206.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

4. Applied egg-rr 2.7%

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

5. Final simplification 2.7%

   \[\leadsto 0.25 \]
6. Add Preprocessing

Alternative 9: 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 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * 0.0d0
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * 0.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

4. Applied egg-rr 3.4%

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

5. Final simplification 3.4%

   \[\leadsto 0 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))