Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 63.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  x
  (fma y (* y (fma (* y y) 0.019444444444444445 0.16666666666666666)) 1.0)))

C

double code(double x, double y) {
	return x / fma(y, (y * fma((y * y), 0.019444444444444445, 0.16666666666666666)), 1.0);
}

Julia

function code(x, y)
	return Float64(x / fma(y, Float64(y * fma(Float64(y * y), 0.019444444444444445, 0.16666666666666666)), 1.0))
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-lowering-sin.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1} \]

   3. associate-*l* N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)} + 1} \]

   4. *-commutative N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot y\right)} + 1} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot y, 1\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}, 1\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}, 1\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

   12. *-lowering-*.f64 64.6

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

7. Simplified 64.6%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]
8. Add Preprocessing

Alternative 3: 57.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y \cdot y\right) \cdot -0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1)
   (* x (fma y (* y -0.16666666666666666) 1.0))
   (/ x (* (* y y) -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1) {
		tmp = x * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = x / ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1)
		tmp = Float64(x * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(x / Float64(Float64(y * y) * -0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1], N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000009

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 71.7

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

   5. Simplified 71.7%

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

   if 3.10000000000000009 < y

   1. Initial program 99.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 23.8

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

   7. Simplified 23.8%

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

   9. Applied egg-rr 24.0%

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

   10. Taylor expanded in y around inf

       \[\leadsto \frac{x}{\color{blue}{\frac{-1}{6} \cdot {y}^{2}}} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{-1}{6} \cdot {y}^{2}}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 24.0

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

   12. Simplified 24.0%

       \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666 \cdot \left(y \cdot y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y \cdot y\right) \cdot -0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{6}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e+36)
   (* x (fma y (* y -0.16666666666666666) 1.0))
   (* x (/ 6.0 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+36) {
		tmp = x * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = x * (6.0 / (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e+36)
		tmp = Float64(x * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(x * Float64(6.0 / Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e+36], N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(6.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000002e36

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   if 5.5000000000000002e36 < y

   1. Initial program 99.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sin-lowering-sin.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 25.4

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

   7. Simplified 25.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{6 \cdot \frac{x}{{y}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{6 \cdot x}{{y}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{{y}^{2}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{6}{{y}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{6}{{y}^{2}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{6}{{y}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto x \cdot \frac{6}{\color{blue}{y \cdot y}} \]

      7. *-lowering-*.f64 25.4

         \[\leadsto x \cdot \frac{6}{\color{blue}{y \cdot y}} \]

   10. Simplified 25.4%

       \[\leadsto \color{blue}{x \cdot \frac{6}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.6e+36)
   (* x (fma y (* y -0.16666666666666666) 1.0))
   (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.6e+36) {
		tmp = x * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.6e+36)
		tmp = Float64(x * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.6e+36], N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5999999999999997e36

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   if 3.5999999999999997e36 < y

   1. Initial program 99.6%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. /-lowering-/.f64 N/A

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

      9. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 23.6%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-lowering-sin.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 64.5

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

7. Simplified 64.5%

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

Alternative 7: 52.1% accurate, 117.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Simplified 55.2%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))