Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.0% → 99.8%

Time: 10.3s

Alternatives: 12

Speedup: 1.5×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 91.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

7. Applied rewrites 94.6%

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

Alternative 3: 91.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

7. Applied rewrites 94.6%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 94.4%

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

Alternative 4: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ t_1 := \left(\left(\mathsf{fma}\left(-0.16666666666666666, t\_0, \frac{\frac{t\_0}{x}}{x}\right) \cdot x\right) \cdot x\right) \cdot y\\ t_2 := \frac{\left(t\_0 \cdot x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -250:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 520:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0))
        (t_1 (* (* (* (fma -0.16666666666666666 t_0 (/ (/ t_0 x) x)) x) x) y))
        (t_2 (/ (* (* t_0 x) y) x)))
   (if (<= y -8.2e+60)
     t_2
     (if (<= y -250.0)
       t_1
       (if (<= y 520.0) (* (/ (sin x) x) y) (if (<= y 2e+61) t_1 t_2))))))

C

double code(double x, double y) {
	double t_0 = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	double t_1 = ((fma(-0.16666666666666666, t_0, ((t_0 / x) / x)) * x) * x) * y;
	double t_2 = ((t_0 * x) * y) / x;
	double tmp;
	if (y <= -8.2e+60) {
		tmp = t_2;
	} else if (y <= -250.0) {
		tmp = t_1;
	} else if (y <= 520.0) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 2e+61) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)
	t_1 = Float64(Float64(Float64(fma(-0.16666666666666666, t_0, Float64(Float64(t_0 / x) / x)) * x) * x) * y)
	t_2 = Float64(Float64(Float64(t_0 * x) * y) / x)
	tmp = 0.0
	if (y <= -8.2e+60)
		tmp = t_2;
	elseif (y <= -250.0)
		tmp = t_1;
	elseif (y <= 520.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 2e+61)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.16666666666666666 * t$95$0 + N[(N[(t$95$0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -8.2e+60], t$95$2, If[LessEqual[y, -250.0], t$95$1, If[LessEqual[y, 520.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2e+61], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2e60 or 1.9999999999999999e61 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.8%

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

   if -8.2e60 < y < -250 or 520 < y < 1.9999999999999999e61

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 76.8%

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

   if -250 < y < 520

   1. Initial program 75.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 5: 84.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

7. Applied rewrites 94.6%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 89.7%

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

Alternative 6: 58.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+201}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, y\right) \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.4e+64)
   (*
    (*
     (fma -0.16666666666666666 (* x x) 1.0)
     (fma (fma (* 0.008333333333333333 y) y 0.16666666666666666) (* y y) 1.0))
    y)
   (if (<= x 1.3e+201)
     (/
      (*
       (fma
        (* y (fma 0.008333333333333333 (* x x) -0.16666666666666666))
        (* x x)
        y)
       x)
      x)
     (/ 0.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.4e+64) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * fma(fma((0.008333333333333333 * y), y, 0.16666666666666666), (y * y), 1.0)) * y;
	} else if (x <= 1.3e+201) {
		tmp = (fma((y * fma(0.008333333333333333, (x * x), -0.16666666666666666)), (x * x), y) * x) / x;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.4e+64)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	elseif (x <= 1.3e+201)
		tmp = Float64(Float64(fma(Float64(y * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), Float64(x * x), y) * x) / x);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.4e+64], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.3e+201], N[(N[(N[(N[(y * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.40000000000000058e64

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.5%

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

   10. Applied rewrites 68.5%

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

   if 9.40000000000000058e64 < x < 1.29999999999999993e201

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.9%

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

   if 1.29999999999999993e201 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.6%

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

Alternative 7: 58.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.4e+64)
   (*
    (*
     (fma -0.16666666666666666 (* x x) 1.0)
     (fma (fma (* 0.008333333333333333 y) y 0.16666666666666666) (* y y) 1.0))
    y)
   (if (<= x 1.3e+201)
     (fma
      (* y (fma 0.008333333333333333 (* x x) -0.16666666666666666))
      (* x x)
      y)
     (/ 0.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.4e+64) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * fma(fma((0.008333333333333333 * y), y, 0.16666666666666666), (y * y), 1.0)) * y;
	} else if (x <= 1.3e+201) {
		tmp = fma((y * fma(0.008333333333333333, (x * x), -0.16666666666666666)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.4e+64)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	elseif (x <= 1.3e+201)
		tmp = fma(Float64(y * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.4e+64], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.3e+201], N[(N[(y * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.40000000000000058e64

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.5%

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

   10. Applied rewrites 68.5%

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

   if 9.40000000000000058e64 < x < 1.29999999999999993e201

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.9%

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

   if 1.29999999999999993e201 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.6%

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

Alternative 8: 57.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.3e+200)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.3e+200) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.3e+200)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.3e+200], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.29999999999999994e200

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

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

   if 5.29999999999999994e200 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.6%

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

Alternative 9: 38.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.3e+201)
   (fma
    (* y (fma 0.008333333333333333 (* x x) -0.16666666666666666))
    (* x x)
    y)
   (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.3e+201) {
		tmp = fma((y * fma(0.008333333333333333, (x * x), -0.16666666666666666)), (x * x), y);
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.3e+201)
		tmp = fma(Float64(y * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), Float64(x * x), y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.3e+201], N[(N[(y * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999993e201

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 43.9%

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

   if 1.29999999999999993e201 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.6%

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

Alternative 10: 37.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7e+62) (* (fma -0.16666666666666666 (* x x) 1.0) y) (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7e+62) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7e+62)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7e+62], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999967e62

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.1%

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

   if 6.99999999999999967e62 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.7%

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

   7. Applied rewrites 40.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 41.0%

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

Alternative 11: 33.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+15}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.1e+15) (* 1.0 y) (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e+15) {
		tmp = 1.0 * y;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.1d+15) then
        tmp = 1.0d0 * y
    else
        tmp = 0.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e+15) {
		tmp = 1.0 * y;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e+15:
		tmp = 1.0 * y
	else:
		tmp = 0.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e+15)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e+15)
		tmp = 1.0 * y;
	else
		tmp = 0.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e+15], N[(1.0 * y), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e15

   1. Initial program 83.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.0%

      \[\leadsto 1 \cdot y \]

   if 1.1e15 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.5%

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

Alternative 12: 27.6% accurate, 36.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 57.7

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

5. Applied rewrites 57.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 33.9%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))