Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

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: 61.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.05:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 0.05)
   (* y (/ x y))
   (* x (fma y (* y -0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 0.05) {
		tmp = y * (x / y);
	} else {
		tmp = x * fma(y, (y * -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.05)
		tmp = Float64(y * Float64(x / y));
	else
		tmp = Float64(x * fma(y, Float64(y * -0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.05], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.050000000000000003

   1. Initial program 99.5%

      \[x \cdot \frac{\sin y}{y} \]
   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 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.5%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[x \cdot \frac{\sin y}{y} \]
   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 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.05:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  x
  (fma
   y
   (*
    y
    (fma
     (* y y)
     (fma y (* y 0.00205026455026455) 0.019444444444444445)
     0.16666666666666666))
   1.0)))

C

double code(double x, double y) {
	return x / fma(y, (y * fma((y * y), fma(y, (y * 0.00205026455026455), 0.019444444444444445), 0.16666666666666666)), 1.0);
}

Julia

function code(x, y)
	return Float64(x / fma(y, Float64(y * fma(Float64(y * y), fma(y, Float64(y * 0.00205026455026455), 0.019444444444444445), 0.16666666666666666)), 1.0))
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.00205026455026455), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] + 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 99.4

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right) + 1}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right) + 1} \]

   3. associate-*l* N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right)\right)} + 1} \]

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

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right), 1\right)}} \]

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

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right)\right)}, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)} \]

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

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)} \]

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

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{7}{360} + \frac{31}{15120} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{31}{15120} \cdot {y}^{2} + \frac{7}{360}}, \frac{1}{6}\right), 1\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{31}{15120}} + \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{31}{15120} + \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

   13. associate-*l* N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \frac{31}{15120}\right)} + \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

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

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{31}{15120}, \frac{7}{360}\right)}, \frac{1}{6}\right), 1\right)} \]

   15. *-lowering-*.f64 66.5

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.00205026455026455}, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)} \]

7. Simplified 66.5%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)}} \]
8. Add Preprocessing

Alternative 4: 62.4% 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 99.4

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}} \]
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. accelerator-lowering-fma.f64 N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 66.5

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

7. Simplified 66.5%

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

Alternative 5: 55.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(x \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.35) (fma -0.16666666666666666 (* y (* x y)) x) (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.35) {
		tmp = fma(-0.16666666666666666, (y * (x * y)), x);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.35)
		tmp = fma(-0.16666666666666666, Float64(y * Float64(x * y)), x);
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.35], N[(-0.16666666666666666 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.35000000000000009

   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. associate-/r/ N/A

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

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

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

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

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

      5. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 72.8

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

   7. Simplified 72.8%

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

   if 2.35000000000000009 < 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 19.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(x \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.5% 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 99.4

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-lowering-*.f64 66.3

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

7. Simplified 66.3%

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

Alternative 7: 50.7% 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 56.4%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))