Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 15

Speedup: 1.0×

• 48.9% 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: 80.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (sin x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (* y y) 0.16666666666666666 1.0)
      (fma (* (* x x) x) -0.16666666666666666 x))
     (if (<= t_1 1.0)
       (*
        (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 t_1 = t_0 * sin(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * fma(((x * x) * x), -0.16666666666666666, x);
	} else if (t_1 <= 1.0) {
		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)
	t_1 = Float64(t_0 * sin(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * fma(Float64(Float64(x * x) * x), -0.16666666666666666, x));
	elseif (t_1 <= 1.0)
		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]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], 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 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 48.5

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

   5. Applied rewrites 48.5%

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

   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 48.2

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

   8. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 48.2%

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

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

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 55.2

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 82.5%

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

Alternative 3: 80.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (sin x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (* y y) 0.16666666666666666 1.0)
      (fma (* (* x x) x) -0.16666666666666666 x))
     (if (<= t_1 1.0)
       (* (fma (* 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 t_1 = t_0 * sin(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * fma(((x * x) * x), -0.16666666666666666, x);
	} else if (t_1 <= 1.0) {
		tmp = fma((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)
	t_1 = Float64(t_0 * sin(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * fma(Float64(Float64(x * x) * x), -0.16666666666666666, x));
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(Float64(0.16666666666666666 * 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]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * 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 48.5

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

   5. Applied rewrites 48.5%

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

   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 48.2

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

   8. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 48.2%

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

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

   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 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 55.2

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 82.4%

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

Alternative 4: 80.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (sin x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (* y y) 0.16666666666666666 1.0)
      (fma (* (* x x) x) -0.16666666666666666 x))
     (if (<= t_1 1.0) (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((y * y), 0.16666666666666666, 1.0) * fma(((x * x) * x), -0.16666666666666666, x);
	} else if (t_1 <= 1.0) {
		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 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * fma(Float64(Float64(x * x) * x), -0.16666666666666666, x));
	elseif (t_1 <= 1.0)
		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[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], 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

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   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 48.2

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

   8. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 48.2%

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

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

   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 98.0

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

   5. Applied rewrites 98.0%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 55.2

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 81.9%

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

Alternative 5: 76.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (sin x))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* y y) 0.16666666666666666 1.0)
      (fma (* (* x x) x) -0.16666666666666666 x))
     (if (<= t_0 1.0)
       (sin x)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        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((y * y), 0.16666666666666666, 1.0) * fma(((x * x) * x), -0.16666666666666666, x);
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * 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 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * fma(Float64(Float64(x * x) * x), -0.16666666666666666, x));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * 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[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   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 48.2

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

   8. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 48.2%

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

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

   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 98.0

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

   5. Applied rewrites 98.0%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 55.2

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 44.7%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 61.0%

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

4. Final simplification 78.1%

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

Alternative 6: 40.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \sin x\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.94:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (sin x))))
   (if (<= t_0 -0.02)
     (* (* -0.16666666666666666 (* x x)) x)
     (if (<= t_0 0.94) (* 1.0 x) (* (* (* y y) x) 0.16666666666666666)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * sin(x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (-0.16666666666666666 * (x * x)) * x;
	} else if (t_0 <= 0.94) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (sinh(y) / y) * sin(x)
    if (t_0 <= (-0.02d0)) then
        tmp = ((-0.16666666666666666d0) * (x * x)) * x
    else if (t_0 <= 0.94d0) 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 t_0 = (Math.sinh(y) / y) * Math.sin(x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = (-0.16666666666666666 * (x * x)) * x;
	} else if (t_0 <= 0.94) {
		tmp = 1.0 * x;
	} else {
		tmp = ((y * y) * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sinh(y) / y) * math.sin(x)
	tmp = 0
	if t_0 <= -0.02:
		tmp = (-0.16666666666666666 * (x * x)) * x
	elif t_0 <= 0.94:
		tmp = 1.0 * x
	else:
		tmp = ((y * y) * x) * 0.16666666666666666
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * sin(x))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * x)) * x);
	elseif (t_0 <= 0.94)
		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)
	t_0 = (sinh(y) / y) * sin(x);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = (-0.16666666666666666 * (x * x)) * x;
	elseif (t_0 <= 0.94)
		tmp = 1.0 * x;
	else
		tmp = ((y * y) * x) * 0.16666666666666666;
	end
	tmp_2 = 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, -0.02], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 0.94], N[(1.0 * x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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 40.3

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

   5. Applied rewrites 40.3%

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

   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 15.8%

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

   10. Applied rewrites 15.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 15.0%

       \[\leadsto \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x \]

   if -0.0200000000000000004 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.93999999999999995

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 8.2

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

   7. Applied rewrites 8.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 72.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 72.7%

       \[\leadsto 1 \cdot x \]

   if 0.93999999999999995 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 47.3

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

   7. Applied rewrites 47.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 38.5%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 38.6%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 0.94:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* t_0 (sin x)) 1.0)
     (*
      (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)) <= 1.0) {
		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)) <= 1.0)
		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], 1.0], 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)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 55.2

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 90.4%

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

Alternative 8: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sinh y) y) (sin x)) -0.02)
   (*
    (fma (* y y) 0.16666666666666666 1.0)
    (fma (* (* x x) x) -0.16666666666666666 x))
   (fma
    (* (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y)
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= -0.02) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * fma(((x * x) * x), -0.16666666666666666, x);
	} else {
		tmp = fma(((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y) * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= -0.02)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * fma(Float64(Float64(x * x) * x), -0.16666666666666666, x));
	else
		tmp = fma(Float64(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y) * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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 68.6

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

   5. Applied rewrites 68.6%

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

   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 30.9

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

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.9%

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

   if -0.0200000000000000004 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 24.9

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

   7. Applied rewrites 24.9%

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

   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 63.1%

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

   12. Applied rewrites 64.3%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sinh y) y) (sin x)) -0.02)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (fma
    (* (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y)
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= -0.02) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = fma(((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y) * y), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= -0.02)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = fma(Float64(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y) * y), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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 40.3

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

   5. Applied rewrites 40.3%

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

   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 15.8%

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

   10. Applied rewrites 15.8%

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

   if -0.0200000000000000004 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 24.9

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

   7. Applied rewrites 24.9%

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

   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 63.1%

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

   12. Applied rewrites 64.3%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sinh y) y) (sin x)) -0.02)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (*
    (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= -0.02) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= -0.02)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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 40.3

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

   5. Applied rewrites 40.3%

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

   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 15.8%

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

   10. Applied rewrites 15.8%

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

   if -0.0200000000000000004 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 24.9

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 58.2%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 64.3%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 0.24:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sinh y) y) (sin x)) 0.24)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (* (* (* y y) x) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= 0.24) {
		tmp = fma(-0.16666666666666666, (x * x), 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)) <= 0.24)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(Float64(Float64(y * y) * x) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], 0.24], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 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)) < 0.23999999999999999

   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 65.6

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

   5. Applied rewrites 65.6%

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

   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 46.8%

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

   10. Applied rewrites 46.8%

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

   if 0.23999999999999999 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 39.1

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

   7. Applied rewrites 39.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 32.1%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 32.2%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 0.24:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.008333333333333333, y \cdot y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (fma (* (* (* y y) x) 0.008333333333333333) (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.02) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = fma((((y * y) * x) * 0.008333333333333333), (y * y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.02)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = fma(Float64(Float64(Float64(y * y) * x) * 0.008333333333333333), Float64(y * y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   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 52.1

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

   5. Applied rewrites 52.1%

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

   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 20.0%

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

   10. Applied rewrites 20.0%

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

   if -0.0200000000000000004 < (sin.f64 x)

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 29.5

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

   7. Applied rewrites 29.5%

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

   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 63.6%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.008333333333333333, y \cdot y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.008333333333333333, y \cdot y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (* (fma (* 0.16666666666666666 y) y 1.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.02) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = fma((0.16666666666666666 * y), y, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.02)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * y), y, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   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 52.1

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

   5. Applied rewrites 52.1%

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

   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 20.0%

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

   10. Applied rewrites 20.0%

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

   if -0.0200000000000000004 < (sin.f64 x)

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 29.5

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.6%

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

   12. Applied rewrites 57.6%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02) (* (* -0.16666666666666666 (* x x)) x) (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.02) {
		tmp = (-0.16666666666666666 * (x * x)) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sin(x) <= (-0.02d0)) then
        tmp = ((-0.16666666666666666d0) * (x * x)) * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sin(x) <= -0.02) {
		tmp = (-0.16666666666666666 * (x * x)) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sin(x) <= -0.02:
		tmp = (-0.16666666666666666 * (x * x)) * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.02)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * x)) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sin(x) <= -0.02)
		tmp = (-0.16666666666666666 * (x * x)) * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   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 52.1

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

   5. Applied rewrites 52.1%

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

   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 20.0%

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

   10. Applied rewrites 20.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 19.2%

       \[\leadsto \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x \]

   if -0.0200000000000000004 < (sin.f64 x)

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   6. 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 29.5

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 57.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 37.7%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.6% 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 91.3

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

4. Applied rewrites 91.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
6. 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 28.2

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

7. Applied rewrites 28.2%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.8%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 27.8%

    \[\leadsto 1 \cdot x \]

14. Final simplification 27.8%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))