Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.3% → 99.6%

Time: 10.4s

Alternatives: 21

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Initial Program: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.3333333333333333\right) \cdot \left(\sinh y\_m \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 1\right)\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (* (* (* x x) -0.3333333333333333) (* (sinh y_m) 0.5))
      (if (<= t_0 2e-8)
        (* (* (/ (sin x) x) (fma (* y_m y_m) 0.16666666666666666 1.0)) y_m)
        (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((x * x) * -0.3333333333333333) * (sinh(y_m) * 0.5);
	} else if (t_0 <= 2e-8) {
		tmp = ((sin(x) / x) * fma((y_m * y_m), 0.16666666666666666, 1.0)) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(x * x) * -0.3333333333333333) * Float64(sinh(y_m) * 0.5));
	elseif (t_0 <= 2e-8)
		tmp = Float64(Float64(Float64(sin(x) / x) * fma(Float64(y_m * y_m), 0.16666666666666666, 1.0)) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(N[Sinh[y$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot \sinh y}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

      4. *-lft-identity N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 \cdot \frac{\sinh y}{x}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\frac{2}{2}} \cdot \frac{\sinh y}{x}\right) \]

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{2}{x} \cdot \frac{\sinh y}{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \frac{\sinh y}{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \frac{\sinh y}{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right)} \cdot \frac{\sinh y}{2} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sin x \cdot \color{blue}{\frac{2}{x}}\right) \cdot \frac{\sinh y}{2} \]

      13. div-inv N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\sinh y \cdot \frac{1}{2}\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sinh y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sinh y\right)} \]

      16. metadata-eval 100.0

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(2 + \frac{-1}{3} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot {x}^{2} + 2\right)} \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {x}^{2}, 2\right)} \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot x}, 2\right) \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]

      4. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{x \cdot x}, 2\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 2\right)} \cdot \left(0.5 \cdot \sinh y\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\frac{-1}{3} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 12.1%

       \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 82.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.6

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right) \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}} + \frac{\sin x}{x}\right) \cdot y \]

      3. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\left({y}^{2} \cdot \frac{\sin x}{x}\right)} \cdot \frac{1}{6} + \frac{\sin x}{x}\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} + \frac{\sin x}{x}\right) \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)} + \frac{\sin x}{x}\right) \cdot y \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 73.5

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.5%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.3333333333333333\right) \cdot \left(\sinh y \cdot 0.5\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.3333333333333333\right) \cdot \left(\sinh y\_m \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (* (* (* x x) -0.3333333333333333) (* (sinh y_m) 0.5))
      (if (<= t_0 2e-8) (* (/ (sin x) x) y_m) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((x * x) * -0.3333333333333333) * (sinh(y_m) * 0.5);
	} else if (t_0 <= 2e-8) {
		tmp = (sin(x) / x) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double t_0 = (Math.sinh(y_m) * Math.sin(x)) / x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((x * x) * -0.3333333333333333) * (Math.sinh(y_m) * 0.5);
	} else if (t_0 <= 2e-8) {
		tmp = (Math.sin(x) / x) * y_m;
	} else {
		tmp = Math.sinh(y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	t_0 = (math.sinh(y_m) * math.sin(x)) / x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((x * x) * -0.3333333333333333) * (math.sinh(y_m) * 0.5)
	elif t_0 <= 2e-8:
		tmp = (math.sin(x) / x) * y_m
	else:
		tmp = math.sinh(y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(x * x) * -0.3333333333333333) * Float64(sinh(y_m) * 0.5));
	elseif (t_0 <= 2e-8)
		tmp = Float64(Float64(sin(x) / x) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	t_0 = (sinh(y_m) * sin(x)) / x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((x * x) * -0.3333333333333333) * (sinh(y_m) * 0.5);
	elseif (t_0 <= 2e-8)
		tmp = (sin(x) / x) * y_m;
	else
		tmp = sinh(y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(N[Sinh[y$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot \sinh y}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

      4. *-lft-identity N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 \cdot \frac{\sinh y}{x}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\frac{2}{2}} \cdot \frac{\sinh y}{x}\right) \]

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{2}{x} \cdot \frac{\sinh y}{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \frac{\sinh y}{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \frac{\sinh y}{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right)} \cdot \frac{\sinh y}{2} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\sin x \cdot \color{blue}{\frac{2}{x}}\right) \cdot \frac{\sinh y}{2} \]

      13. div-inv N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\sinh y \cdot \frac{1}{2}\right)} \]

      14. *-commutative N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sinh y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sinh y\right)} \]

      16. metadata-eval 100.0

          \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(2 + \frac{-1}{3} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot {x}^{2} + 2\right)} \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {x}^{2}, 2\right)} \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot x}, 2\right) \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]

      4. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{x \cdot x}, 2\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 2\right)} \cdot \left(0.5 \cdot \sinh y\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\frac{-1}{3} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \sinh y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 12.1%

       \[\leadsto \left(-0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 82.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.6

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 73.5

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.5%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.3333333333333333\right) \cdot \left(\sinh y \cdot 0.5\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma
         (fma (* -0.0001984126984126984 (* x x)) (* x x) -0.16666666666666666)
         (* x x)
         1.0))
       y_m)
      (if (<= t_0 2e-8) (* (/ (sin x) x) y_m) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma(fma((-0.0001984126984126984 * (x * x)), (x * x), -0.16666666666666666), (x * x), 1.0)) * y_m;
	} else if (t_0 <= 2e-8) {
		tmp = (sin(x) / x) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(fma(Float64(-0.0001984126984126984 * Float64(x * x)), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y_m);
	elseif (t_0 <= 2e-8)
		tmp = Float64(Float64(sin(x) / x) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040} \cdot {x}^{2}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 54.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 82.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.6

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 73.5

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.5%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma
         (fma (* -0.0001984126984126984 (* x x)) (* x x) -0.16666666666666666)
         (* x x)
         1.0))
       y_m)
      (if (<= t_0 2e-66) (* (/ y_m x) (sin x)) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma(fma((-0.0001984126984126984 * (x * x)), (x * x), -0.16666666666666666), (x * x), 1.0)) * y_m;
	} else if (t_0 <= 2e-66) {
		tmp = (y_m / x) * sin(x);
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(fma(Float64(-0.0001984126984126984 * Float64(x * x)), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y_m);
	elseif (t_0 <= 2e-66)
		tmp = Float64(Float64(y_m / x) * sin(x));
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e-66], N[(N[(y$95$m / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040} \cdot {x}^{2}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 54.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-66

   1. Initial program 81.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 99.5

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.4%

      \[\leadsto \frac{y}{x} \cdot \color{blue}{\sin x} \]

   if 2e-66 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 65.7

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.6%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x} \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-116)
      (*
       (*
        (fma
         (*
          (fma
           (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
           (* x x)
           -0.16666666666666666)
          x)
         x
         1.0)
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0))
       y_m)
      (if (<= t_0 1e-279) (* (- 1.0 (- 1.0 y_m)) 0.5) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (fma((fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666) * x), x, 1.0) * fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0)) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(Float64(fma(Float64(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666) * x), x, 1.0) * fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0)) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-116

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 59.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -2e-116 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 74.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 47.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.6%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y\_m \cdot y\_m, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-116)
      (*
       (*
        (fma
         (*
          (fma
           (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
           (* x x)
           -0.16666666666666666)
          x)
         x
         1.0)
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0))
       y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y_m y_m) 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)
         0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (fma((fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666) * x), x, 1.0) * fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0)) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = (fma(fma(fma(0.0003968253968253968, (y_m * y_m), 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(Float64(fma(Float64(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666) * x), x, 1.0) * fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0)) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y_m * y_m), 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-116

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 59.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -2e-116 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 74.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 47.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y\_m \cdot y\_m, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-116)
      (*
       (*
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0)
        (fma
         (fma (* -0.0001984126984126984 (* x x)) (* x x) -0.16666666666666666)
         (* x x)
         1.0))
       y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y_m y_m) 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)
         0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * fma(fma((-0.0001984126984126984 * (x * x)), (x * x), -0.16666666666666666), (x * x), 1.0)) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = (fma(fma(fma(0.0003968253968253968, (y_m * y_m), 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * fma(fma(Float64(-0.0001984126984126984 * Float64(x * x)), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y_m * y_m), 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-116

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040} \cdot {x}^{2}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 58.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -2e-116 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 74.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 47.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-207}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right)\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y\_m \cdot y\_m, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-207)
      (*
       (*
        (fma (* x x) -0.16666666666666666 1.0)
        (fma
         (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
         (* y_m y_m)
         1.0))
       y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y_m y_m) 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)
         0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-207) {
		tmp = (fma((x * x), -0.16666666666666666, 1.0) * fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0)) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = (fma(fma(fma(0.0003968253968253968, (y_m * y_m), 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-207)
		tmp = Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0)) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y_m * y_m), 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-207], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-207

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 56.4%

      \[\leadsto \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]

   if -1.99999999999999985e-207 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 72.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.2%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-207}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y\_m \cdot y\_m, 0.016666666666666666\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-116)
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y_m y_m) 0.016666666666666666)
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)
         0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = (fma(fma(fma(0.0003968253968253968, (y_m * y_m), 0.016666666666666666), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y_m * y_m), 0.016666666666666666), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-116

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 17.3

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 23.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y \]

   if -2e-116 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 74.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 47.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968 \cdot \left(y\_m \cdot y\_m\right), y\_m \cdot y\_m, 0.3333333333333333\right), y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-116)
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (*
          (fma
           (fma
            (* 0.0003968253968253968 (* y_m y_m))
            (* y_m y_m)
            0.3333333333333333)
           (* y_m y_m)
           2.0)
          y_m)
         0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = (fma(fma((0.0003968253968253968 * (y_m * y_m)), (y_m * y_m), 0.3333333333333333), (y_m * y_m), 2.0) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(Float64(0.0003968253968253968 * Float64(y_m * y_m)), Float64(y_m * y_m), 0.3333333333333333), Float64(y_m * y_m), 2.0) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-116

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 17.3

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 23.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y \]

   if -2e-116 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 74.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 47.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520} \cdot {y}^{2}, y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968 \cdot \left(y \cdot y\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968 \cdot \left(y \cdot y\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-116)
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (fma
          (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
          (* y_m y_m)
          1.0)
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-116

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 17.3

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 23.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y \]

   if -2e-116 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 74.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 47.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 63.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y\_m \cdot y\_m, 0.16666666666666666\right), y\_m \cdot y\_m, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-207)
      (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (*
         (fma
          (fma 0.008333333333333333 (* y_m y_m) 0.16666666666666666)
          (* y_m y_m)
          1.0)
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-207) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = fma(fma(0.008333333333333333, (y_m * y_m), 0.16666666666666666), (y_m * y_m), 1.0) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-207)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y_m * y_m), 0.16666666666666666), Float64(y_m * y_m), 1.0) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-207], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-207

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 24.1

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -1.99999999999999985e-207 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 72.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.2%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 63.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -2e-207)
      (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
      (if (<= t_0 1e-279)
        (* (- 1.0 (- 1.0 y_m)) 0.5)
        (* (* (fma 0.3333333333333333 (* y_m y_m) 2.0) y_m) 0.5))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-207) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else if (t_0 <= 1e-279) {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	} else {
		tmp = (fma(0.3333333333333333, (y_m * y_m), 2.0) * y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-207)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	elseif (t_0 <= 1e-279)
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	else
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y_m * y_m), 2.0) * y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-207], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1e-279], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-207

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 24.1

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if -1.99999999999999985e-207 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000006e-279

   1. Initial program 72.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.2%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]

   if 1.00000000000000006e-279 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 50.1

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-279}:\\ \;\;\;\;\left(1 - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(\left(\sinh y\_m \cdot 0.5\right) \cdot \left(\frac{2}{x} \cdot \sin x\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (* (* (sinh y_m) 0.5) (* (/ 2.0 x) (sin x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * ((sinh(y_m) * 0.5) * ((2.0 / x) * sin(x)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * ((sinh(y_m) * 0.5d0) * ((2.0d0 / x) * sin(x)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * ((Math.sinh(y_m) * 0.5) * ((2.0 / x) * Math.sin(x)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * ((math.sinh(y_m) * 0.5) * ((2.0 / x) * math.sin(x)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(Float64(sinh(y_m) * 0.5) * Float64(Float64(2.0 / x) * sin(x))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * ((sinh(y_m) * 0.5) * ((2.0 / x) * sin(x)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(N[(N[Sinh[y$95$m], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(2.0 / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sinh y}}{x} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   4. *-lft-identity N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(1 \cdot \frac{\sinh y}{x}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sin x \cdot \left(\color{blue}{\frac{2}{2}} \cdot \frac{\sinh y}{x}\right) \]

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(\frac{2}{x} \cdot \frac{\sinh y}{2}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \frac{\sinh y}{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \frac{\sinh y}{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right)} \cdot \frac{\sinh y}{2} \]

   12. lower-/.f64 N/A

       \[\leadsto \left(\sin x \cdot \color{blue}{\frac{2}{x}}\right) \cdot \frac{\sinh y}{2} \]

   13. div-inv N/A

       \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\sinh y \cdot \frac{1}{2}\right)} \]

   14. *-commutative N/A

       \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sinh y\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sinh y\right)} \]

   16. metadata-eval 99.9

       \[\leadsto \left(\sin x \cdot \frac{2}{x}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{x}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

5. Final simplification 99.9%

   \[\leadsto \left(\sinh y \cdot 0.5\right) \cdot \left(\frac{2}{x} \cdot \sin x\right) \]
6. Add Preprocessing

Alternative 15: 99.8% accurate, 1.0× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* (/ (sinh y_m) x) (sin x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * ((sinh(y_m) / x) * sin(x));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * ((sinh(y_m) / x) * sin(x))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * ((Math.sinh(y_m) / x) * Math.sin(x));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * ((math.sinh(y_m) / x) * math.sin(x))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(Float64(sinh(y_m) / x) * sin(x)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * ((sinh(y_m) / x) * sin(x));
end

Wolfram

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

Derivation

1. Initial program 91.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sinh y}}{x} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   6. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]
5. Add Preprocessing

Alternative 16: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.36 \cdot 10^{+31}:\\ \;\;\;\;\sinh y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(e^{y\_m} - 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (if (<= x 1.36e+31) (sinh y_m) (* (- (exp y_m) 1.0) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.36e+31) {
		tmp = sinh(y_m);
	} else {
		tmp = (exp(y_m) - 1.0) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (x <= 1.36d+31) then
        tmp = sinh(y_m)
    else
        tmp = (exp(y_m) - 1.0d0) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.36e+31) {
		tmp = Math.sinh(y_m);
	} else {
		tmp = (Math.exp(y_m) - 1.0) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if x <= 1.36e+31:
		tmp = math.sinh(y_m)
	else:
		tmp = (math.exp(y_m) - 1.0) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= 1.36e+31)
		tmp = sinh(y_m);
	else
		tmp = Float64(Float64(exp(y_m) - 1.0) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (x <= 1.36e+31)
		tmp = sinh(y_m);
	else
		tmp = (exp(y_m) - 1.0) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, 1.36e+31], N[Sinh[y$95$m], $MachinePrecision], N[(N[(N[Exp[y$95$m], $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3600000000000001e31

   1. Initial program 88.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 54.8

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.1%

      \[\leadsto \sinh y \]

   if 1.3600000000000001e31 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 61.6

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(e^{y} - 1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.0%

      \[\leadsto \left(e^{y} - 1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 10:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y\_m, -1\right), y\_m, 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= x 10.0)
    (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
    (* (- 1.0 (fma (fma 0.5 y_m -1.0) y_m 1.0)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 10.0) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else {
		tmp = (1.0 - fma(fma(0.5, y_m, -1.0), y_m, 1.0)) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= 10.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	else
		tmp = Float64(Float64(1.0 - fma(fma(0.5, y_m, -1.0), y_m, 1.0)) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, 10.0], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 * y$95$m + -1.0), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 10

   1. Initial program 88.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 52.7

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 38.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 38.2%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if 10 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 58.2

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.3%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.1%

       \[\leadsto \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= x 1.22e+54)
    (* (fma (* -0.16666666666666666 x) x 1.0) y_m)
    (* (- 1.0 (- 1.0 y_m)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.22e+54) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * y_m;
	} else {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= 1.22e+54)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * y_m);
	else
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, 1.22e+54], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.22e54

   1. Initial program 88.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 51.7

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 37.1%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot y \]

   if 1.22e54 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 65.0

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 37.7%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 34.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.36 \cdot 10^{+31}:\\ \;\;\;\;1 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(1 - y\_m\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (if (<= x 1.36e+31) (* 1.0 y_m) (* (- 1.0 (- 1.0 y_m)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.36e+31) {
		tmp = 1.0 * y_m;
	} else {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (x <= 1.36d+31) then
        tmp = 1.0d0 * y_m
    else
        tmp = (1.0d0 - (1.0d0 - y_m)) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.36e+31) {
		tmp = 1.0 * y_m;
	} else {
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if x <= 1.36e+31:
		tmp = 1.0 * y_m
	else:
		tmp = (1.0 - (1.0 - y_m)) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= 1.36e+31)
		tmp = Float64(1.0 * y_m);
	else
		tmp = Float64(Float64(1.0 - Float64(1.0 - y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (x <= 1.36e+31)
		tmp = 1.0 * y_m;
	else
		tmp = (1.0 - (1.0 - y_m)) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, 1.36e+31], N[(1.0 * y$95$m), $MachinePrecision], N[(N[(1.0 - N[(1.0 - y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3600000000000001e31

   1. Initial program 88.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 51.9

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 33.7%

      \[\leadsto 1 \cdot y \]

   if 1.3600000000000001e31 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 61.6

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.8%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - \left(1 + -1 \cdot y\right)\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.8%

       \[\leadsto \left(1 - \left(1 - y\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 33.9% accurate, 14.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;1 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (if (<= x 1.15e+20) (* 1.0 y_m) (* (- 1.0 1.0) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.15e+20) {
		tmp = 1.0 * y_m;
	} else {
		tmp = (1.0 - 1.0) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (x <= 1.15d+20) then
        tmp = 1.0d0 * y_m
    else
        tmp = (1.0d0 - 1.0d0) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.15e+20) {
		tmp = 1.0 * y_m;
	} else {
		tmp = (1.0 - 1.0) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if x <= 1.15e+20:
		tmp = 1.0 * y_m
	else:
		tmp = (1.0 - 1.0) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= 1.15e+20)
		tmp = Float64(1.0 * y_m);
	else
		tmp = Float64(Float64(1.0 - 1.0) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (x <= 1.15e+20)
		tmp = 1.0 * y_m;
	else
		tmp = (1.0 - 1.0) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, 1.15e+20], N[(1.0 * y$95$m), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e20

   1. Initial program 88.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

      5. lower-sin.f64 52.7

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 34.2%

      \[\leadsto 1 \cdot y \]

   if 1.15e20 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      6. lower-exp.f64 N/A

         \[\leadsto \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \cdot \frac{1}{2} \]

      7. lower-neg.f64 58.4

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(1 - e^{-y}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.1%

      \[\leadsto \left(1 - e^{-y}\right) \cdot 0.5 \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 - 1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.3%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 28.2% accurate, 36.2× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* 1.0 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (1.0 * y_m);
}

Fortran

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

Java

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

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * (1.0 * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(1.0 * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * (1.0 * y_m);
end

Wolfram

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

Derivation

1. Initial program 91.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

   5. lower-sin.f64 52.8

      \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto 1 \cdot y \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))