Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.9%

Time: 10.6s

Alternatives: 18

Speedup: 1.0×

• 49.9% 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: 89.1% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-12}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1e-12) (sinh y) (/ (* (sin x_m) (sinh y)) x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1e-12) {
		tmp = sinh(y);
	} else {
		tmp = (sin(x_m) * sinh(y)) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 1d-12) then
        tmp = sinh(y)
    else
        tmp = (sin(x_m) * sinh(y)) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1e-12) {
		tmp = Math.sinh(y);
	} else {
		tmp = (Math.sin(x_m) * Math.sinh(y)) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 1e-12:
		tmp = math.sinh(y)
	else:
		tmp = (math.sin(x_m) * math.sinh(y)) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1e-12)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(sin(x_m) * sinh(y)) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 1e-12)
		tmp = sinh(y);
	else
		tmp = (sin(x_m) * sinh(y)) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1e-12], N[Sinh[y], $MachinePrecision], N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999998e-13

   1. Initial program 89.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 75.2%

      \[\leadsto \sinh y \]

   if 9.9999999999999998e-13 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-35}:\\ \;\;\;\;\left(\frac{\sin x\_m}{x\_m} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (*
         (fma
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         y)
        y
        1.0))
      y)
     (if (<= t_0 1e-35)
       (*
        (*
         (/ (sin x_m) x_m)
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0))
        y)
       (sinh y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0)) * y;
	} else if (t_0 <= 1e-35) {
		tmp = ((sin(x_m) / x_m) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0)) * y);
	elseif (t_0 <= 1e-35)
		tmp = Float64(Float64(Float64(sin(x_m) / x_m) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-35], N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000001e-35

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   if 1.00000000000000001e-35 < (/.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 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 79.5%

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

Alternative 3: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-35}:\\ \;\;\;\;\left(\frac{\sin x\_m}{x\_m} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (*
         (fma
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         y)
        y
        1.0))
      y)
     (if (<= t_0 1e-35)
       (* (* (/ (sin x_m) x_m) (fma 0.16666666666666666 (* y y) 1.0)) y)
       (sinh y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0)) * y;
	} else if (t_0 <= 1e-35) {
		tmp = ((sin(x_m) / x_m) * fma(0.16666666666666666, (y * y), 1.0)) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0)) * y);
	elseif (t_0 <= 1e-35)
		tmp = Float64(Float64(Float64(sin(x_m) / x_m) * fma(0.16666666666666666, Float64(y * y), 1.0)) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-35], N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000001e-35

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.5%

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

   if 1.00000000000000001e-35 < (/.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 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 79.5%

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

Alternative 4: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-35}:\\ \;\;\;\;\frac{\sin x\_m}{x\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (*
         (fma
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         y)
        y
        1.0))
      y)
     (if (<= t_0 1e-35) (* (/ (sin x_m) x_m) y) (sinh y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0)) * y;
	} else if (t_0 <= 1e-35) {
		tmp = (sin(x_m) / x_m) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0)) * y);
	elseif (t_0 <= 1e-35)
		tmp = Float64(Float64(sin(x_m) / x_m) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-35], N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000001e-35

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.00000000000000001e-35 < (/.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 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 79.5%

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

Alternative 5: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x\_m \cdot x\_m, 0.16666666666666666\right), x\_m \cdot x\_m, 1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (*
         (fma
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         y)
        y
        1.0))
      y)
     (if (<= t_0 1e-59)
       (*
        (/
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0)
         (fma
          (fma 0.019444444444444445 (* x_m x_m) 0.16666666666666666)
          (* x_m x_m)
          1.0))
        y)
       (sinh y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0)) * y;
	} else if (t_0 <= 1e-59) {
		tmp = (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / fma(fma(0.019444444444444445, (x_m * x_m), 0.16666666666666666), (x_m * x_m), 1.0)) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0)) * y);
	elseif (t_0 <= 1e-59)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / fma(fma(0.019444444444444445, Float64(x_m * x_m), 0.16666666666666666), Float64(x_m * x_m), 1.0)) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-59], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(0.019444444444444445 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-59

   1. Initial program 84.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)} \cdot y \]

   if 1e-59 < (/.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 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 80.0%

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

Alternative 6: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\\ t_1 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot t\_0\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x\_m \cdot x\_m, 0.16666666666666666\right), x\_m \cdot x\_m, 1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0
         (fma
          (*
           (fma
            (fma (* y y) 0.0001984126984126984 0.008333333333333333)
            (* y y)
            0.16666666666666666)
           y)
          y
          1.0))
        (t_1 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_1 (- INFINITY))
     (* (* (fma -0.16666666666666666 (* x_m x_m) 1.0) t_0) y)
     (if (<= t_1 1e-59)
       (*
        (/
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0)
         (fma
          (fma 0.019444444444444445 (* x_m x_m) 0.16666666666666666)
          (* x_m x_m)
          1.0))
        y)
       (* t_0 y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0);
	double t_1 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * t_0) * y;
	} else if (t_1 <= 1e-59) {
		tmp = (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / fma(fma(0.019444444444444445, (x_m * x_m), 0.16666666666666666), (x_m * x_m), 1.0)) * y;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0)
	t_1 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * t_0) * y);
	elseif (t_1 <= 1e-59)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / fma(fma(0.019444444444444445, Float64(x_m * x_m), 0.16666666666666666), Float64(x_m * x_m), 1.0)) * y);
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e-59], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(0.019444444444444445 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(t$95$0 * y), $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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-59

   1. Initial program 84.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, x \cdot x, 0.16666666666666666\right), x \cdot x, 1\right)} \cdot y \]

   if 1e-59 < (/.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 y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 71.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, 0.16666666666666666, 1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968 \cdot \left(y \cdot y\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (*
         (fma
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         y)
        y
        1.0))
      y)
     (if (<= t_0 0.005)
       (*
        (/
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0)
         (fma (* x_m x_m) 0.16666666666666666 1.0))
        y)
       (*
        (*
         (fma
          (fma (* 0.0003968253968253968 (* y y)) (* y y) 0.3333333333333333)
          (* y y)
          2.0)
         y)
        0.5)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0)) * y;
	} else if (t_0 <= 0.005) {
		tmp = (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / fma((x_m * x_m), 0.16666666666666666, 1.0)) * y;
	} else {
		tmp = (fma(fma((0.0003968253968253968 * (y * y)), (y * y), 0.3333333333333333), (y * y), 2.0) * y) * 0.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0)) * y);
	elseif (t_0 <= 0.005)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / fma(Float64(x_m * x_m), 0.16666666666666666, 1.0)) * y);
	else
		tmp = Float64(Float64(fma(fma(Float64(0.0003968253968253968 * Float64(y * y)), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * 0.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * 0.5), $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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0050000000000000001

   1. Initial program 85.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 71.1%

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

   if 0.0050000000000000001 < (/.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 77.9

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

   5. Applied rewrites 77.9%

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

   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 68.2%

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

   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 68.2%

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

Alternative 8: 69.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{\mathsf{fma}\left(x\_m \cdot x\_m, 0.16666666666666666, 1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968 \cdot \left(y \cdot y\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y y)
        1.0))
      y)
     (if (<= t_0 0.005)
       (*
        (/
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0)
         (fma (* x_m x_m) 0.16666666666666666 1.0))
        y)
       (*
        (*
         (fma
          (fma (* 0.0003968253968253968 (* y y)) (* y y) 0.3333333333333333)
          (* y y)
          2.0)
         y)
        0.5)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else if (t_0 <= 0.005) {
		tmp = (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / fma((x_m * x_m), 0.16666666666666666, 1.0)) * y;
	} else {
		tmp = (fma(fma((0.0003968253968253968 * (y * y)), (y * y), 0.3333333333333333), (y * y), 2.0) * y) * 0.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	elseif (t_0 <= 0.005)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / fma(Float64(x_m * x_m), 0.16666666666666666, 1.0)) * y);
	else
		tmp = Float64(Float64(fma(fma(Float64(0.0003968253968253968 * Float64(y * y)), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * 0.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * 0.5), $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 74.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0050000000000000001

   1. Initial program 85.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 71.1%

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

   if 0.0050000000000000001 < (/.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 77.9

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

   5. Applied rewrites 77.9%

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

   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 68.2%

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

   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 68.2%

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

Alternative 9: 57.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 -2e-163)
     (*
      (*
       (fma -0.16666666666666666 (* x_m x_m) 1.0)
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y y)
        1.0))
      y)
     (if (<= t_0 1e-302)
       (* (- (+ 1.0 y) (- 1.0 y)) 0.5)
       (*
        (fma
         (*
          (fma
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           (* y y)
           0.16666666666666666)
          y)
         y
         1.0)
        y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -2e-163) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else if (t_0 <= 1e-302) {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	} else {
		tmp = fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= -2e-163)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	elseif (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-163], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-302], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-163

   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 82.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.0%

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

   if -1.99999999999999985e-163 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999996e-303

   1. Initial program 72.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.5%

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

   if 9.9999999999999996e-303 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x\_m \cdot x\_m, 0.008333333333333333\right), x\_m \cdot x\_m, -0.16666666666666666\right), x\_m \cdot x\_m, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 -1e-143)
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* x_m x_m) 0.008333333333333333)
        (* x_m x_m)
        -0.16666666666666666)
       (* x_m x_m)
       1.0)
      y)
     (if (<= t_0 1e-302)
       (* (- (+ 1.0 y) (- 1.0 y)) 0.5)
       (*
        (fma
         (*
          (fma
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           (* y y)
           0.16666666666666666)
          y)
         y
         1.0)
        y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -1e-143) {
		tmp = fma(fma(fma(-0.0001984126984126984, (x_m * x_m), 0.008333333333333333), (x_m * x_m), -0.16666666666666666), (x_m * x_m), 1.0) * y;
	} else if (t_0 <= 1e-302) {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	} else {
		tmp = fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= -1e-143)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(x_m * x_m), 0.008333333333333333), Float64(x_m * x_m), -0.16666666666666666), Float64(x_m * x_m), 1.0) * y);
	elseif (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-143], N[(N[(N[(N[(-0.0001984126984126984 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-302], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999995e-144

   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 28.7

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

   5. Applied rewrites 28.7%

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

   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 32.7%

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

   if -9.9999999999999995e-144 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999996e-303

   1. Initial program 73.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 48.3%

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

   if 9.9999999999999996e-303 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 -2e-163)
     (*
      (*
       (fma (* x_m x_m) -0.16666666666666666 1.0)
       (fma 0.16666666666666666 (* y y) 1.0))
      y)
     (if (<= t_0 1e-302)
       (* (- (+ 1.0 y) (- 1.0 y)) 0.5)
       (*
        (fma
         (*
          (fma
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           (* y y)
           0.16666666666666666)
          y)
         y
         1.0)
        y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -2e-163) {
		tmp = (fma((x_m * x_m), -0.16666666666666666, 1.0) * fma(0.16666666666666666, (y * y), 1.0)) * y;
	} else if (t_0 <= 1e-302) {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	} else {
		tmp = fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= -2e-163)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), -0.16666666666666666, 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0)) * y);
	elseif (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-163], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-302], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-163

   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 82.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.7%

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

   if -1.99999999999999985e-163 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999996e-303

   1. Initial program 72.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.5%

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

   if 9.9999999999999996e-303 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x} + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{1}{120} \cdot \frac{\sin x}{x}\right)\right) + \frac{\sin x}{x}\right) \cdot y} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{\sin x}{x} \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 -2e-163)
     (*
      (*
       (fma (* x_m x_m) -0.16666666666666666 1.0)
       (fma 0.16666666666666666 (* y y) 1.0))
      y)
     (if (<= t_0 1e-302)
       (* (- (+ 1.0 y) (- 1.0 y)) 0.5)
       (*
        (fma
         (fma (* y y) 0.008333333333333333 0.16666666666666666)
         (* y y)
         1.0)
        y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -2e-163) {
		tmp = (fma((x_m * x_m), -0.16666666666666666, 1.0) * fma(0.16666666666666666, (y * y), 1.0)) * y;
	} else if (t_0 <= 1e-302) {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= -2e-163)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), -0.16666666666666666, 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0)) * y);
	elseif (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	else
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-163], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-302], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-163

   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 82.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.7%

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

   if -1.99999999999999985e-163 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999996e-303

   1. Initial program 72.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.5%

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

   if 9.9999999999999996e-303 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   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 88.3%

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

   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 57.7%

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

Alternative 13: 45.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 -2e-163)
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) y)
     (if (<= t_0 1e-302)
       (* (- (+ 1.0 y) (- 1.0 y)) 0.5)
       (*
        (fma
         (fma (* y y) 0.008333333333333333 0.16666666666666666)
         (* y y)
         1.0)
        y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -2e-163) {
		tmp = fma(-0.16666666666666666, (x_m * x_m), 1.0) * y;
	} else if (t_0 <= 1e-302) {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= -2e-163)
		tmp = Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * y);
	elseif (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	else
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-163], N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-302], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-163

   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 30.4

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

   5. Applied rewrites 30.4%

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

   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 28.5%

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

   if -1.99999999999999985e-163 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999996e-303

   1. Initial program 72.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.5%

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

   if 9.9999999999999996e-303 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   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 88.3%

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

   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 57.7%

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

Alternative 14: 43.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sin x\_m \cdot \sinh y}{x\_m}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x_m) (sinh y)) x_m)))
   (if (<= t_0 -2e-163)
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) y)
     (if (<= t_0 1e-302)
       (* (- (+ 1.0 y) (- 1.0 y)) 0.5)
       (* (* (fma 0.3333333333333333 (* y y) 2.0) y) 0.5)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (sin(x_m) * sinh(y)) / x_m;
	double tmp;
	if (t_0 <= -2e-163) {
		tmp = fma(-0.16666666666666666, (x_m * x_m), 1.0) * y;
	} else if (t_0 <= 1e-302) {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	} else {
		tmp = (fma(0.3333333333333333, (y * y), 2.0) * y) * 0.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(sin(x_m) * sinh(y)) / x_m)
	tmp = 0.0
	if (t_0 <= -2e-163)
		tmp = Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * y);
	elseif (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	else
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y * y), 2.0) * y) * 0.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-163], N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-302], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999985e-163

   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 30.4

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

   5. Applied rewrites 30.4%

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

   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 28.5%

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

   if -1.99999999999999985e-163 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999996e-303

   1. Initial program 72.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.5%

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

   if 9.9999999999999996e-303 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   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 52.3%

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

Alternative 15: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{-1}{\frac{x\_m}{\sin x\_m}} \cdot \left(-\sinh y\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (* (/ -1.0 (/ x_m (sin x_m))) (- (sinh y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return (-1.0 / (x_m / sin(x_m))) * -sinh(y);
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = ((-1.0d0) / (x_m / sin(x_m))) * -sinh(y)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return (-1.0 / (x_m / Math.sin(x_m))) * -Math.sinh(y);
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return (-1.0 / (x_m / math.sin(x_m))) * -math.sinh(y)

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(Float64(-1.0 / Float64(x_m / sin(x_m))) * Float64(-sinh(y)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y)
	tmp = (-1.0 / (x_m / sin(x_m))) * -sinh(y);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(N[(-1.0 / N[(x$95$m / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sinh[y], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 92.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sin x \cdot \sinh y}}} \]

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. distribute-neg-frac2 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{-1}{\frac{x}{\sin x}} \cdot \left(-\sinh y\right)} \]
5. Add Preprocessing

Alternative 16: 38.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 6.2 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 6.2e+147)
   (* (fma -0.16666666666666666 (* x_m x_m) 1.0) y)
   (* (- (+ 1.0 y) (- 1.0 y)) 0.5)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 6.2e+147) {
		tmp = fma(-0.16666666666666666, (x_m * x_m), 1.0) * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 6.2e+147)
		tmp = Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 6.2e+147], N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2000000000000001e147

   1. Initial program 91.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   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 33.6%

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

   if 6.2000000000000001e147 < 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 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.2%

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

Alternative 17: 38.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+27}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + y\right) - \left(1 - y\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1e+27) (* 1.0 y) (* (- (+ 1.0 y) (- 1.0 y)) 0.5)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1e+27) {
		tmp = 1.0 * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 1d+27) then
        tmp = 1.0d0 * y
    else
        tmp = ((1.0d0 + y) - (1.0d0 - y)) * 0.5d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1e+27) {
		tmp = 1.0 * y;
	} else {
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 1e+27:
		tmp = 1.0 * y
	else:
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1e+27)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(Float64(Float64(1.0 + y) - Float64(1.0 - y)) * 0.5);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 1e+27)
		tmp = 1.0 * y;
	else
		tmp = ((1.0 + y) - (1.0 - y)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1e+27], N[(1.0 * y), $MachinePrecision], N[(N[(N[(1.0 + y), $MachinePrecision] - N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e27

   1. Initial program 89.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 32.9%

       \[\leadsto 1 \cdot y \]

   if 1e27 < 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 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 25.3%

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

Alternative 18: 27.1% accurate, 36.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \cdot y \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (* 1.0 y))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return 1.0 * y;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = 1.0d0 * y
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return 1.0 * y;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return 1.0 * y

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(1.0 * y)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y)
	tmp = 1.0 * y;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 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

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

8. Applied rewrites 36.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 25.6%

    \[\leadsto 1 \cdot y \]
12. Add Preprocessing

Developer Target 1: 99.9% 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 2024324 
(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))