Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 19

Speedup: 1.0×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \sin x\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+132}:\\ \;\;\;\;t\_2 \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))
        (t_1 (* t_0 (sin x)))
        (t_2
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 (fma (pow x 3.0) -0.16666666666666666 x))
     (if (<= t_1 1e+132) (* t_2 (sin x)) (* t_0 x)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * sin(x);
	double t_2 = fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * fma(pow(x, 3.0), -0.16666666666666666, x);
	} else if (t_1 <= 1e+132) {
		tmp = t_2 * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * sin(x))
	t_2 = fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * fma((x ^ 3.0), -0.16666666666666666, x));
	elseif (t_1 <= 1e+132)
		tmp = Float64(t_2 * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+132], N[(t$95$2 * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

      9. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, \frac{-1}{6}, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{-1}{6}, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval 68.3

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \]

   8. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

      9. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \sin x\\ t_2 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+132}:\\ \;\;\;\;t\_2 \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))
        (t_1 (* t_0 (sin x)))
        (t_2 (fma (* y y) 0.16666666666666666 1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 (fma (pow x 3.0) -0.16666666666666666 x))
     (if (<= t_1 1e+132) (* t_2 (sin x)) (* t_0 x)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * sin(x);
	double t_2 = fma((y * y), 0.16666666666666666, 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * fma(pow(x, 3.0), -0.16666666666666666, x);
	} else if (t_1 <= 1e+132) {
		tmp = t_2 * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * sin(x))
	t_2 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * fma((x ^ 3.0), -0.16666666666666666, x));
	elseif (t_1 <= 1e+132)
		tmp = Float64(t_2 * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+132], N[(t$95$2 * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{-1}{6}, x\right)} \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{-1}{6}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right) \]

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 50.4

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

   8. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \sin x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+132}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (sin x))))
   (if (<= t_1 (- INFINITY))
     (fma (pow x 3.0) -0.16666666666666666 x)
     (if (<= t_1 1e+132) (sin x) (* t_0 x)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * sin(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else if (t_1 <= 1e+132) {
		tmp = sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * sin(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma((x ^ 3.0), -0.16666666666666666, x);
	elseif (t_1 <= 1e+132)
		tmp = sin(x);
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+132], N[Sin[x], $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 2.7

         \[\leadsto \color{blue}{\sin x} \]

   5. Applied rewrites 2.7%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 16.7%

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{-0.16666666666666666}, x\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 98.1

         \[\leadsto \color{blue}{\sin x} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\sin x} \]

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \sin x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (sin x))))
   (if (<= t_0 (- INFINITY))
     (fma (pow x 3.0) -0.16666666666666666 x)
     (if (<= t_0 1.0)
       (sin x)
       (*
        (*
         (/ 0.5 y)
         (*
          (fma
           (fma
            (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
            (* y y)
            0.3333333333333333)
           (* y y)
           2.0)
          y))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * sin(x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = ((0.5 / y) * (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * sin(x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma((x ^ 3.0), -0.16666666666666666, x);
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(Float64(Float64(0.5 / y) * Float64(fma(fma(fma(0.0003968253968253968, Float64(y * y), 0.016666666666666666), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(N[(N[(0.5 / y), $MachinePrecision] * N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 2.7

         \[\leadsto \color{blue}{\sin x} \]

   5. Applied rewrites 2.7%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 16.7%

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{-0.16666666666666666}, x\right) \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 98.1

         \[\leadsto \color{blue}{\sin x} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\sin x} \]

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 62.7%

      \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \left(2 \cdot \sinh y\right)}}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.8%

       \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot \color{blue}{y}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 79.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* t_0 (sin x)) 1e+132)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      (sin x))
     (* t_0 x))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((t_0 * sin(x)) <= 1e+132) {
		tmp = fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(t_0 * sin(x)) <= 1e+132)
		tmp = Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], 1e+132], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

      14. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* t_0 (sin x)) 1e+132)
     (*
      (fma
       (fma (* (* y y) 0.0001984126984126984) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      (sin x))
     (* t_0 x))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((t_0 * sin(x)) <= 1e+132) {
		tmp = fma(fma(((y * y) * 0.0001984126984126984), (y * y), 0.16666666666666666), (y * y), 1.0) * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(t_0 * sin(x)) <= 1e+132)
		tmp = Float64(fma(fma(Float64(Float64(y * y) * 0.0001984126984126984), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], 1e+132], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

      14. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.9%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* t_0 (sin x)) 1e+132)
     (*
      (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
      (sin x))
     (* t_0 x))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((t_0 * sin(x)) <= 1e+132) {
		tmp = fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(t_0 * sin(x)) <= 1e+132)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], 1e+132], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* t_0 (sin x)) 1e+132)
     (* (fma (* 0.008333333333333333 (* y y)) (* y y) 1.0) (sin x))
     (* t_0 x))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((t_0 * sin(x)) <= 1e+132) {
		tmp = fma((0.008333333333333333 * (y * y)), (y * y), 1.0) * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(t_0 * sin(x)) <= 1e+132)
		tmp = Float64(fma(Float64(0.008333333333333333 * Float64(y * y)), Float64(y * y), 1.0) * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], 1e+132], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 92.0%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* t_0 (sin x)) 1e+132)
     (* (fma (* y y) 0.16666666666666666 1.0) (sin x))
     (* t_0 x))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((t_0 * sin(x)) <= 1e+132) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * sin(x);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(t_0 * sin(x)) <= 1e+132)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], 1e+132], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.99999999999999991e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if 9.99999999999999991e131 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 84.7%

      \[\leadsto \frac{\sinh y}{y} \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sinh y) y) (sin x)) 1.0)
   (sin x)
   (*
    (*
     (/ 0.5 y)
     (*
      (fma
       (fma
        (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
        (* y y)
        0.3333333333333333)
       (* y y)
       2.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = ((0.5 / y) * (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(Float64(Float64(0.5 / y) * Float64(fma(fma(fma(0.0003968253968253968, Float64(y * y), 0.016666666666666666), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], 1.0], N[Sin[x], $MachinePrecision], N[(N[(N[(0.5 / y), $MachinePrecision] * N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 61.8

         \[\leadsto \color{blue}{\sin x} \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\sin x} \]

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 62.7%

      \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \left(2 \cdot \sinh y\right)}}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.8%

       \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot \color{blue}{y}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 79.8%

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

4. Final simplification 65.9%

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

Alternative 12: 36.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 0.98:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sinh y) y) (sin x)) 0.98)
   (* 1.0 x)
   (* (* (* y y) x) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= 0.98) {
		tmp = 1.0 * x;
	} else {
		tmp = ((y * y) * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) / y) * sin(x)) <= 0.98d0) then
        tmp = 1.0d0 * x
    else
        tmp = ((y * y) * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sinh(y) / y) * Math.sin(x)) <= 0.98) {
		tmp = 1.0 * x;
	} else {
		tmp = ((y * y) * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) / y) * math.sin(x)) <= 0.98:
		tmp = 1.0 * x
	else:
		tmp = ((y * y) * x) * 0.16666666666666666
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= 0.98)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(y * y) * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) / y) * sin(x)) <= 0.98)
		tmp = 1.0 * x;
	else
		tmp = ((y * y) * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], 0.98], N[(1.0 * x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.97999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 24.4

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

   5. Applied rewrites 24.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 34.3%

       \[\leadsto 1 \cdot x \]

   if 0.97999999999999998 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 48.4%

       \[\leadsto \left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 0.98:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.5}{y} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (*
   (/ 0.5 y)
   (*
    (fma
     (fma
      (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
      (* y y)
      0.3333333333333333)
     (* y y)
     2.0)
    y))
  x))

C

double code(double x, double y) {
	return ((0.5 / y) * (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y)) * x;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(0.5 / y) * Float64(fma(fma(fma(0.0003968253968253968, Float64(y * y), 0.016666666666666666), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y)) * x)
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 / y), $MachinePrecision] * N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 33.2

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

5. Applied rewrites 33.2%

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

7. Applied rewrites 56.6%

   \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \left(2 \cdot \sinh y\right)}}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 52.6%

    \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot \color{blue}{y}\right) \]
11. Step-by-step derivation

12. Applied rewrites 62.2%

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

13. Final simplification 62.2%

    \[\leadsto \left(\frac{0.5}{y} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right)\right) \cdot x \]
14. Add Preprocessing

Alternative 14: 58.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (fma
   (fma
    (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
    (* y y)
    0.16666666666666666)
   (* y y)
   1.0)
  x))

C

double code(double x, double y) {
	return fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * x;
}

Julia

function code(x, y)
	return Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * x)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l/ N/A

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

   6. +-lft-identity N/A

      \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

   7. flip-+ N/A

      \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \color{blue}{\frac{0 \cdot 0 - \left(\mathsf{neg}\left(\sinh y\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}{0 - \left(\mathsf{neg}\left(\sinh y\right)\right)}} \]

   8. sqr-neg N/A

      \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \frac{0 \cdot 0 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sinh y\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sinh y\right)\right)\right)\right)}}{0 - \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

   9. remove-double-neg N/A

      \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \frac{0 \cdot 0 - \color{blue}{\sinh y} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sinh y\right)\right)\right)\right)}{0 - \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

   10. remove-double-neg N/A

       \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \frac{0 \cdot 0 - \sinh y \cdot \color{blue}{\sinh y}}{0 - \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

   11. neg-sub0 N/A

       \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \frac{0 \cdot 0 - \sinh y \cdot \sinh y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sinh y\right)\right)\right)}} \]

   12. remove-double-neg N/A

       \[\leadsto \frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \frac{0 \cdot 0 - \sinh y \cdot \sinh y}{\color{blue}{\sinh y}} \]

   13. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \left(0 \cdot 0 - \sinh y \cdot \sinh y\right)}{\sinh y}} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\sin x}{\mathsf{neg}\left(y\right)} \cdot \left(0 \cdot 0 - \sinh y \cdot \sinh y\right)}{\sinh y}} \]

4. Applied rewrites 23.9%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 91.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 61.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{x} \]
11. Add Preprocessing

Alternative 15: 56.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot x, y \cdot y, x\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma
  (*
   (fma
    (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
    (* y y)
    0.16666666666666666)
   x)
  (* y y)
  x))

C

double code(double x, double y) {
	return fma((fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666) * x), (y * y), x);
}

Julia

function code(x, y)
	return fma(Float64(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666) * x), Float64(y * y), x)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 33.2

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

5. Applied rewrites 33.2%

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

7. Applied rewrites 56.6%

   \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \left(2 \cdot \sinh y\right)}}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 60.4%

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

11. Final simplification 60.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot x, y \cdot y, x\right) \]
12. Add Preprocessing

Alternative 16: 55.9% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 33.2

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

5. Applied rewrites 33.2%

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

7. Applied rewrites 56.6%

   \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \left(2 \cdot \sinh y\right)}}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 59.5%

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

12. Applied rewrites 59.5%

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

13. Final simplification 59.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot x \]
14. Add Preprocessing

Alternative 17: 55.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 33.2

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

5. Applied rewrites 33.2%

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

7. Applied rewrites 56.6%

   \[\leadsto \left(\frac{0.5}{y} \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \left(2 \cdot \sinh y\right)}}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 59.5%

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

11. Taylor expanded in y around inf

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

13. Applied rewrites 59.4%

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

14. Final simplification 59.4%

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

Alternative 18: 47.9% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 33.2

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

5. Applied rewrites 33.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 48.3%

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

10. Applied rewrites 48.3%

    \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot x \]
11. Add Preprocessing

Alternative 19: 26.2% accurate, 36.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 33.2

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

5. Applied rewrites 33.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 48.3%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 26.9%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))