Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.5% → 99.8%

Time: 12.6s

Alternatives: 18

Speedup: 1.5×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right) \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left|0.008333333333333333 \cdot y\right|, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -0.005)
     (/
      (*
       (fma
        (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)
        x
        x)
       y)
      x)
     (if (<= t_0 1e-80)
       (* (/ (sin x) x) y)
       (*
        (fma
         (fma (fabs (* 0.008333333333333333 y)) y 0.16666666666666666)
         (* y y)
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -0.005) {
		tmp = (fma(((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y), x, x) * y) / x;
	} else if (t_0 <= 1e-80) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = fma(fma(fabs((0.008333333333333333 * y)), y, 0.16666666666666666), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(Float64(fma(Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y), x, x) * y) / x);
	elseif (t_0 <= 1e-80)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(fma(fma(abs(Float64(0.008333333333333333 * y)), y, 0.16666666666666666), Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-80], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[Abs[N[(0.008333333333333333 * y), $MachinePrecision]], $MachinePrecision] * y + 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) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.8%

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

   if -0.0050000000000000001 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999961e-81

   1. Initial program 77.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 9.99999999999999961e-81 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 76.3%

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

   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 62.6%

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

   10. Applied rewrites 80.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right) \cdot y}{x}\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-80}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left|0.008333333333333333 \cdot y\right|, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{y} - e^{-y}\right) \cdot 0.5\\ t_1 := \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ t_2 := \frac{t\_1 \cdot y}{x}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\frac{t\_1}{x} \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (exp y) (exp (- y))) 0.5))
        (t_1
         (*
          (sin x)
          (fma
           (fma (* y y) 0.008333333333333333 0.16666666666666666)
           (* y y)
           1.0)))
        (t_2 (/ (* t_1 y) x)))
   (if (<= y -1.35e+62)
     t_2
     (if (<= y -1.35)
       t_0
       (if (<= y 0.5) (* (/ t_1 x) y) (if (<= y 1.3e+52) t_0 t_2))))))

C

double code(double x, double y) {
	double t_0 = (exp(y) - exp(-y)) * 0.5;
	double t_1 = sin(x) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	double t_2 = (t_1 * y) / x;
	double tmp;
	if (y <= -1.35e+62) {
		tmp = t_2;
	} else if (y <= -1.35) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = (t_1 / x) * y;
	} else if (y <= 1.3e+52) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(exp(y) - exp(Float64(-y))) * 0.5)
	t_1 = Float64(sin(x) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0))
	t_2 = Float64(Float64(t_1 * y) / x)
	tmp = 0.0
	if (y <= -1.35e+62)
		tmp = t_2;
	elseif (y <= -1.35)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = Float64(Float64(t_1 / x) * y);
	elseif (y <= 1.3e+52)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Exp[y], $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -1.35e+62], t$95$2, If[LessEqual[y, -1.35], t$95$0, If[LessEqual[y, 0.5], N[(N[(t$95$1 / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.3e+52], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e62 or 1.3e52 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

   if -1.35e62 < y < -1.3500000000000001 or 0.5 < y < 1.3e52

   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 93.1

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

   5. Applied rewrites 93.1%

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

   if -1.3500000000000001 < y < 0.5

   1. Initial program 79.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

Alternative 4: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot \left(y \cdot y\right)\\ t_1 := \frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x}{x} \cdot y\\ t_2 := -0.008333333333333333 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.027777777777777776 - t\_2 \cdot t\_2}{0.16666666666666666 + t\_2}, y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;y \leq -0.0014:\\ \;\;\;\;\frac{1 - t\_0 \cdot t\_0}{1 - t\_0} \cdot y\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{x} \cdot \sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+136}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y)))
        (t_1 (* (/ (* (fma (* y y) 0.16666666666666666 1.0) (sin x)) x) y))
        (t_2 (* -0.008333333333333333 (* y y))))
   (if (<= y -6.5e+152)
     t_1
     (if (<= y -1e+79)
       (*
        (fma
         (/ (- 0.027777777777777776 (* t_2 t_2)) (+ 0.16666666666666666 t_2))
         (* y y)
         1.0)
        y)
       (if (<= y -0.0014)
         (* (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0)) y)
         (if (<= y 8.6e+49)
           (* (/ y x) (sin x))
           (if (<= y 7e+136)
             (*
              (*
               (fma -0.16666666666666666 (* x x) 1.0)
               (fma
                (fma (* y y) 0.008333333333333333 0.16666666666666666)
                (* y y)
                1.0))
              y)
             t_1)))))))

C

double code(double x, double y) {
	double t_0 = fma(0.008333333333333333, (y * y), 0.16666666666666666) * (y * y);
	double t_1 = ((fma((y * y), 0.16666666666666666, 1.0) * sin(x)) / x) * y;
	double t_2 = -0.008333333333333333 * (y * y);
	double tmp;
	if (y <= -6.5e+152) {
		tmp = t_1;
	} else if (y <= -1e+79) {
		tmp = fma(((0.027777777777777776 - (t_2 * t_2)) / (0.16666666666666666 + t_2)), (y * y), 1.0) * y;
	} else if (y <= -0.0014) {
		tmp = ((1.0 - (t_0 * t_0)) / (1.0 - t_0)) * y;
	} else if (y <= 8.6e+49) {
		tmp = (y / x) * sin(x);
	} else if (y <= 7e+136) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * Float64(y * y))
	t_1 = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x)) / x) * y)
	t_2 = Float64(-0.008333333333333333 * Float64(y * y))
	tmp = 0.0
	if (y <= -6.5e+152)
		tmp = t_1;
	elseif (y <= -1e+79)
		tmp = Float64(fma(Float64(Float64(0.027777777777777776 - Float64(t_2 * t_2)) / Float64(0.16666666666666666 + t_2)), Float64(y * y), 1.0) * y);
	elseif (y <= -0.0014)
		tmp = Float64(Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0)) * y);
	elseif (y <= 8.6e+49)
		tmp = Float64(Float64(y / x) * sin(x));
	elseif (y <= 7e+136)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(-0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+152], t$95$1, If[LessEqual[y, -1e+79], N[(N[(N[(N[(0.027777777777777776 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, -0.0014], N[(N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 8.6e+49], N[(N[(y / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+136], N[(N[(N[(-0.16666666666666666 * N[(x * x), $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], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -6.4999999999999997e152 or 7.00000000000000002e136 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.6%

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

   if -6.4999999999999997e152 < y < -9.99999999999999967e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   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 94.7%

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

   10. Applied rewrites 94.7%

       \[\leadsto \mathsf{fma}\left(\frac{0.027777777777777776 - \left(-0.008333333333333333 \cdot \left(y \cdot y\right)\right) \cdot \left(-0.008333333333333333 \cdot \left(y \cdot y\right)\right)}{0.16666666666666666 + -0.008333333333333333 \cdot \left(y \cdot y\right)}, y \cdot y, 1\right) \cdot y \]

   if -9.99999999999999967e78 < y < -0.00139999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 10.2%

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

   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 10.7%

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

   10. Applied rewrites 54.5%

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

   if -0.00139999999999999999 < y < 8.5999999999999998e49

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 91.5

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

   7. Applied rewrites 91.5%

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

   if 8.5999999999999998e49 < y < 7.00000000000000002e136

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.0%

      \[\leadsto \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.027777777777777776 - \left(-0.008333333333333333 \cdot \left(y \cdot y\right)\right) \cdot \left(-0.008333333333333333 \cdot \left(y \cdot y\right)\right)}{0.16666666666666666 + -0.008333333333333333 \cdot \left(y \cdot y\right)}, y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;y \leq -0.0014:\\ \;\;\;\;\frac{1 - \left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot \left(y \cdot y\right)\right)}{1 - \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot \left(y \cdot y\right)} \cdot y\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{x} \cdot \sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+136}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{if}\;x \leq 2.02 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{t\_0}{x} \cdot y\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin x \cdot t\_0\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0)))
   (if (<= x 2.02e-32)
     (* (* (/ t_0 x) y) (sin x))
     (/ (* (* (sin x) t_0) y) x))))

C

double code(double x, double y) {
	double t_0 = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	double tmp;
	if (x <= 2.02e-32) {
		tmp = ((t_0 / x) * y) * sin(x);
	} else {
		tmp = ((sin(x) * t_0) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)
	tmp = 0.0
	if (x <= 2.02e-32)
		tmp = Float64(Float64(Float64(t_0 / x) * y) * sin(x));
	else
		tmp = Float64(Float64(Float64(sin(x) * t_0) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, 2.02e-32], N[(N[(N[(t$95$0 / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0200000000000001e-32

   1. Initial program 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

   7. Applied rewrites 89.8%

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

   if 2.0200000000000001e-32 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.3%

      \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{+50}:\\ \;\;\;\;\left(\frac{t\_0}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+136}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.16666666666666666 1.0)))
   (if (<= y 7.2e+50)
     (* (* (/ t_0 x) y) (sin x))
     (if (<= y 7e+136)
       (*
        (*
         (fma -0.16666666666666666 (* x x) 1.0)
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0))
        y)
       (* (/ (* t_0 (sin x)) x) y)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0);
	double tmp;
	if (y <= 7.2e+50) {
		tmp = ((t_0 / x) * y) * sin(x);
	} else if (y <= 7e+136) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else {
		tmp = ((t_0 * sin(x)) / x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	tmp = 0.0
	if (y <= 7.2e+50)
		tmp = Float64(Float64(Float64(t_0 / x) * y) * sin(x));
	elseif (y <= 7e+136)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	else
		tmp = Float64(Float64(Float64(t_0 * sin(x)) / x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, If[LessEqual[y, 7.2e+50], N[(N[(N[(t$95$0 / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+136], N[(N[(N[(-0.16666666666666666 * N[(x * x), $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], N[(N[(N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.19999999999999972e50

   1. Initial program 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 87.0%

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

   if 7.19999999999999972e50 < y < 7.00000000000000002e136

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.0%

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

   if 7.00000000000000002e136 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.2%

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

Alternative 7: 91.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (*
   (/
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    x)
   y)
  (sin x)))

C

double code(double x, double y) {
	return ((fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) / x) * y) * sin(x);
}

Julia

function code(x, y)
	return Float64(Float64(Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) / x) * y) * sin(x))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

7. Applied rewrites 90.4%

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

Alternative 8: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)}{x} \cdot y\right) \cdot \sin x \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* (/ (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) x) y) (sin x)))

C

double code(double x, double y) {
	return ((fma(((y * y) * 0.008333333333333333), (y * y), 1.0) / x) * y) * sin(x);
}

Julia

function code(x, y)
	return Float64(Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) / x) * y) * sin(x))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

7. Applied rewrites 90.4%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 90.1%

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

Alternative 9: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{5} \cdot 0.008333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e+16)
   (*
    (*
     (fma -0.16666666666666666 (* x x) 1.0)
     (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
    y)
   (* (pow y 5.0) 0.008333333333333333)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e+16) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	} else {
		tmp = pow(y, 5.0) * 0.008333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e+16)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	else
		tmp = Float64((y ^ 5.0) * 0.008333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e+16], N[(N[(N[(-0.16666666666666666 * N[(x * x), $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], N[(N[Power[y, 5.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5e16

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.6%

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

   if 4.5e16 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left({y}^{3}, \color{blue}{\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)}, y\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{120} \cdot {y}^{\color{blue}{5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.0%

       \[\leadsto {y}^{5} \cdot 0.008333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y \cdot y, y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{elif}\;x \leq 4.47 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.8e+84)
   (fma (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) (* y y) y)
   (if (<= x 5.8e+147)
     (/ (* (* (fma (* -0.16666666666666666 x) x 1.0) x) y) x)
     (if (<= x 4.47e+229)
       (fma
        (* y (fma 0.008333333333333333 (* x x) -0.16666666666666666))
        (* x x)
        y)
       (/ 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.8e+84) {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), (y * y), y);
	} else if (x <= 5.8e+147) {
		tmp = ((fma((-0.16666666666666666 * x), x, 1.0) * x) * y) / x;
	} else if (x <= 4.47e+229) {
		tmp = fma((y * fma(0.008333333333333333, (x * x), -0.16666666666666666)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.8e+84)
		tmp = fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), Float64(y * y), y);
	elseif (x <= 5.8e+147)
		tmp = Float64(Float64(Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x) * y) / x);
	elseif (x <= 4.47e+229)
		tmp = fma(Float64(y * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.8e+84], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 5.8e+147], N[(N[(N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 4.47e+229], N[(N[(y * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.80000000000000032e84

   1. Initial program 87.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.5%

      \[\leadsto \mathsf{fma}\left({y}^{3}, \color{blue}{\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)}, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

   if 7.80000000000000032e84 < x < 5.7999999999999997e147

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.5

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 16.1%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\sin \mathsf{PI}\left(\right) + x \cdot \left(-1 \cdot \cos \mathsf{PI}\left(\right) + x \cdot \left(\frac{-1}{2} \cdot \sin \mathsf{PI}\left(\right) + \frac{1}{6} \cdot \left(x \cdot \cos \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot y}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 36.0%

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

   if 5.7999999999999997e147 < x < 4.46999999999999997e229

   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.6%

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

   if 4.46999999999999997e229 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \frac{0}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y \cdot y, y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.16666666666666666, x \cdot x, y\right)\\ \mathbf{elif}\;x \leq 4.47 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.2e+107)
   (fma (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) (* y y) y)
   (if (<= x 5.8e+147)
     (fma (* y -0.16666666666666666) (* x x) y)
     (if (<= x 4.47e+229)
       (fma
        (* y (fma 0.008333333333333333 (* x x) -0.16666666666666666))
        (* x x)
        y)
       (/ 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.2e+107) {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), (y * y), y);
	} else if (x <= 5.8e+147) {
		tmp = fma((y * -0.16666666666666666), (x * x), y);
	} else if (x <= 4.47e+229) {
		tmp = fma((y * fma(0.008333333333333333, (x * x), -0.16666666666666666)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.2e+107)
		tmp = fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), Float64(y * y), y);
	elseif (x <= 5.8e+147)
		tmp = fma(Float64(y * -0.16666666666666666), Float64(x * x), y);
	elseif (x <= 4.47e+229)
		tmp = fma(Float64(y * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.2e+107], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 5.8e+147], N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 4.47e+229], N[(N[(y * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.2000000000000001e107

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.9%

      \[\leadsto \mathsf{fma}\left({y}^{3}, \color{blue}{\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)}, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.9%

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

   if 9.2000000000000001e107 < x < 5.7999999999999997e147

   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 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.1%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 10.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 37.6%

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

   if 5.7999999999999997e147 < x < 4.46999999999999997e229

   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.6%

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

   if 4.46999999999999997e229 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \frac{0}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.16666666666666666, x \cdot x, y\right)\\ \mathbf{elif}\;x \leq 4.47 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.2e+107)
   (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y)
   (if (<= x 5.8e+147)
     (fma (* y -0.16666666666666666) (* x x) y)
     (if (<= x 4.47e+229)
       (fma
        (* y (fma 0.008333333333333333 (* x x) -0.16666666666666666))
        (* x x)
        y)
       (/ 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.2e+107) {
		tmp = fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y;
	} else if (x <= 5.8e+147) {
		tmp = fma((y * -0.16666666666666666), (x * x), y);
	} else if (x <= 4.47e+229) {
		tmp = fma((y * fma(0.008333333333333333, (x * x), -0.16666666666666666)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.2e+107)
		tmp = Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y);
	elseif (x <= 5.8e+147)
		tmp = fma(Float64(y * -0.16666666666666666), Float64(x * x), y);
	elseif (x <= 4.47e+229)
		tmp = fma(Float64(y * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.2e+107], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 5.8e+147], N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 4.47e+229], N[(N[(y * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.2000000000000001e107

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.9%

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

   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 60.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 60.6%

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

   if 9.2000000000000001e107 < x < 5.7999999999999997e147

   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 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.1%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 10.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 37.6%

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

   if 5.7999999999999997e147 < x < 4.46999999999999997e229

   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.6%

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

   if 4.46999999999999997e229 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \frac{0}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.16666666666666666, x \cdot x, y\right)\\ \mathbf{elif}\;x \leq 4.47 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.2e+107)
   (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y)
   (if (<= x 5.8e+147)
     (fma (* y -0.16666666666666666) (* x x) y)
     (if (<= x 4.47e+229)
       (fma (* y (* (* x x) 0.008333333333333333)) (* x x) y)
       (/ 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.2e+107) {
		tmp = fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y;
	} else if (x <= 5.8e+147) {
		tmp = fma((y * -0.16666666666666666), (x * x), y);
	} else if (x <= 4.47e+229) {
		tmp = fma((y * ((x * x) * 0.008333333333333333)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.2e+107)
		tmp = Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y);
	elseif (x <= 5.8e+147)
		tmp = fma(Float64(y * -0.16666666666666666), Float64(x * x), y);
	elseif (x <= 4.47e+229)
		tmp = fma(Float64(y * Float64(Float64(x * x) * 0.008333333333333333)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.2e+107], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 5.8e+147], N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 4.47e+229], N[(N[(y * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.2000000000000001e107

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.9%

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

   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 60.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 60.6%

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

   if 9.2000000000000001e107 < x < 5.7999999999999997e147

   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 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.1%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 10.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 37.6%

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

   if 5.7999999999999997e147 < x < 4.46999999999999997e229

   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.8%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 35.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 35.6%

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

   if 4.46999999999999997e229 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \frac{0}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.16666666666666666, x \cdot x, y\right)\\ \mathbf{elif}\;x \leq 4.47 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.2e+107)
   (* (fma 0.16666666666666666 (* y y) 1.0) y)
   (if (<= x 5.8e+147)
     (fma (* y -0.16666666666666666) (* x x) y)
     (if (<= x 4.47e+229)
       (fma (* y (* (* x x) 0.008333333333333333)) (* x x) y)
       (/ 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.2e+107) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * y;
	} else if (x <= 5.8e+147) {
		tmp = fma((y * -0.16666666666666666), (x * x), y);
	} else if (x <= 4.47e+229) {
		tmp = fma((y * ((x * x) * 0.008333333333333333)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.2e+107)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * y);
	elseif (x <= 5.8e+147)
		tmp = fma(Float64(y * -0.16666666666666666), Float64(x * x), y);
	elseif (x <= 4.47e+229)
		tmp = fma(Float64(y * Float64(Float64(x * x) * 0.008333333333333333)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.2e+107], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 5.8e+147], N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 4.47e+229], N[(N[(y * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.2000000000000001e107

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.9%

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

   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 60.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 57.5%

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

   if 9.2000000000000001e107 < x < 5.7999999999999997e147

   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 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.1%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 10.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 37.6%

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

   if 5.7999999999999997e147 < x < 4.46999999999999997e229

   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.8%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 35.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 35.6%

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

   if 4.46999999999999997e229 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.3

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

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \frac{0}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.16666666666666666, x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.2e+107)
   (* (fma 0.16666666666666666 (* y y) 1.0) y)
   (if (<= x 9e+228) (fma (* y -0.16666666666666666) (* x x) y) (/ 0.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.2e+107) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * y;
	} else if (x <= 9e+228) {
		tmp = fma((y * -0.16666666666666666), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.2e+107)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * y);
	elseif (x <= 9e+228)
		tmp = fma(Float64(y * -0.16666666666666666), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.2e+107], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 9e+228], N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.2000000000000001e107

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.9%

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

   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 60.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 57.5%

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

   if 9.2000000000000001e107 < x < 8.99999999999999966e228

   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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.5%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 24.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 30.9%

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

   if 8.99999999999999966e228 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 46.2%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.7%

       \[\leadsto \frac{0}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.16666666666666666, x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9e+228) (fma (* y -0.16666666666666666) (* x x) y) (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9e+228) {
		tmp = fma((y * -0.16666666666666666), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9e+228)
		tmp = fma(Float64(y * -0.16666666666666666), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9e+228], N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.99999999999999966e228

   1. Initial program 89.4%

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

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 28.4%

      \[\leadsto 1 \cdot y \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 37.7%

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

   if 8.99999999999999966e228 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 46.2%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.7%

       \[\leadsto \frac{0}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 33.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+55}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 7.4e+55) (* 1.0 y) (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.4e+55) {
		tmp = 1.0 * y;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7.4d+55) then
        tmp = 1.0d0 * y
    else
        tmp = 0.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.4e+55) {
		tmp = 1.0 * y;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.4e+55:
		tmp = 1.0 * y
	else:
		tmp = 0.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.4e+55)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.4e+55)
		tmp = 1.0 * y;
	else
		tmp = 0.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.4e+55], N[(1.0 * y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4000000000000004e55

   1. Initial program 87.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.6%

      \[\leadsto 1 \cdot y \]

   if 7.4000000000000004e55 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 22.9%

      \[\leadsto \frac{\sin \left(\left(-x\right) + \mathsf{PI}\left(\right)\right) \cdot y}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.8%

       \[\leadsto \frac{0}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 28.3% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 y))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 * y;
}

Python

def code(x, y):
	return 1.0 * y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 49.4

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

5. Applied rewrites 49.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.9%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(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))