Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 11.5s

Alternatives: 11

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: 57.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \frac{x}{\mathsf{fma}\left(y, -0.16666666666666666, \frac{-1}{y}\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3200.0)
   (fma
    (* y y)
    (*
     x
     (fma
      (* y y)
      (fma (* y y) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666))
    x)
   (/ (- 0.0 (/ x (fma y -0.16666666666666666 (/ -1.0 y)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3200.0) {
		tmp = fma((y * y), (x * fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x);
	} else {
		tmp = (0.0 - (x / fma(y, -0.16666666666666666, (-1.0 / y)))) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3200.0)
		tmp = fma(Float64(y * y), Float64(x * fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x);
	else
		tmp = Float64(Float64(0.0 - Float64(x / fma(y, -0.16666666666666666, Float64(-1.0 / y)))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3200.0], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.0 - N[(x / N[(y * -0.16666666666666666 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3200

   1. Initial program 99.9%

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

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

      1. +-commutative N/A

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

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

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

   7. Simplified 70.0%

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

   if 3200 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   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. clear-num N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. associate-/r* N/A

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

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

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

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

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

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

      13. sin-lowering-sin.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{\frac{x}{\frac{-1}{\sin y}}}{\color{blue}{0 - y}} \]

      15. --lowering--.f64 99.5

          \[\leadsto \frac{\frac{x}{\frac{-1}{\sin y}}}{\color{blue}{0 - y}} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{-1}{\sin y}}}{0 - y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{\color{blue}{\frac{\frac{-1}{6} \cdot {y}^{2} - 1}{y}}}}{0 - y} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\frac{\frac{-1}{6} \cdot {y}^{2}}{y} - \frac{1}{y}}}}{0 - y} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{x}{\frac{\frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}}{y} - \frac{1}{y}}}{0 - y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot y\right) \cdot y}}{y} - \frac{1}{y}}}{0 - y} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(\frac{-1}{6} \cdot y\right) \cdot \frac{y}{y}} - \frac{1}{y}}}{0 - y} \]

      5. *-inverses N/A

         \[\leadsto \frac{\frac{x}{\left(\frac{-1}{6} \cdot y\right) \cdot \color{blue}{1} - \frac{1}{y}}}{0 - y} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\frac{-1}{6} \cdot y} - \frac{1}{y}}}{0 - y} \]

      7. *-inverses N/A

         \[\leadsto \frac{\frac{x}{\frac{-1}{6} \cdot y - \frac{\color{blue}{\frac{y}{y}}}{y}}}{0 - y} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{\frac{-1}{6} \cdot y - \color{blue}{\frac{y}{y \cdot y}}}}{0 - y} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{x}{\frac{-1}{6} \cdot y - \frac{y}{\color{blue}{{y}^{2}}}}}{0 - y} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{\frac{x}{\frac{-1}{6} \cdot y - \frac{\color{blue}{y \cdot 1}}{{y}^{2}}}}{0 - y} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\frac{x}{\frac{-1}{6} \cdot y - \color{blue}{y \cdot \frac{1}{{y}^{2}}}}}{0 - y} \]

      12. sub-neg N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\frac{-1}{6} \cdot y + \left(\mathsf{neg}\left(y \cdot \frac{1}{{y}^{2}}\right)\right)}}}{0 - y} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(y \cdot \frac{1}{{y}^{2}}\right)\right)}}{0 - y} \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\frac{x}{y \cdot \frac{-1}{6} + \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{1}{{y}^{2}}\right)\right)}}}{0 - y} \]

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

          \[\leadsto \frac{\frac{x}{\color{blue}{\mathsf{fma}\left(y, \frac{-1}{6}, y \cdot \left(\mathsf{neg}\left(\frac{1}{{y}^{2}}\right)\right)\right)}}}{0 - y} \]

      16. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \color{blue}{\mathsf{neg}\left(y \cdot \frac{1}{{y}^{2}}\right)}\right)}}{0 - y} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \mathsf{neg}\left(\color{blue}{\frac{y \cdot 1}{{y}^{2}}}\right)\right)}}{0 - y} \]

      18. *-rgt-identity N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \mathsf{neg}\left(\frac{\color{blue}{y}}{{y}^{2}}\right)\right)}}{0 - y} \]

      19. unpow2 N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \mathsf{neg}\left(\frac{y}{\color{blue}{y \cdot y}}\right)\right)}}{0 - y} \]

      20. associate-/r* N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \mathsf{neg}\left(\color{blue}{\frac{\frac{y}{y}}{y}}\right)\right)}}{0 - y} \]

      21. *-inverses N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \mathsf{neg}\left(\frac{\color{blue}{1}}{y}\right)\right)}}{0 - y} \]

      22. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{y}}\right)}}{0 - y} \]

      23. metadata-eval N/A

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, \frac{-1}{6}, \frac{\color{blue}{-1}}{y}\right)}}{0 - y} \]

      24. /-lowering-/.f64 24.5

          \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(y, -0.16666666666666666, \color{blue}{\frac{-1}{y}}\right)}}{0 - y} \]

   7. Simplified 24.5%

      \[\leadsto \frac{\frac{x}{\color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, \frac{-1}{y}\right)}}}{0 - y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \frac{x}{\mathsf{fma}\left(y, -0.16666666666666666, \frac{-1}{y}\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3200.0)
   (fma
    (* y y)
    (*
     x
     (fma
      (* y y)
      (fma (* y y) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666))
    x)
   (/ 1.0 (/ (/ y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3200.0) {
		tmp = fma((y * y), (x * fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x);
	} else {
		tmp = 1.0 / ((y / x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3200.0)
		tmp = fma(Float64(y * y), Float64(x * fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x);
	else
		tmp = Float64(1.0 / Float64(Float64(y / x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3200.0], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(1.0 / N[(N[(y / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3200

   1. Initial program 99.9%

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

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

      1. +-commutative N/A

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

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

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

   7. Simplified 70.0%

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

   if 3200 < y

   1. Initial program 99.7%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.2

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

   7. Simplified 4.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. clear-num N/A

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

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

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

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

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

      9. /-lowering-/.f64 23.6

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

   9. Applied egg-rr 23.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{-1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 0\right), -1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (/
   -1.0
   (fma
    y
    (fma y (fma y (fma y 0.008333333333333333 0.0) -0.16666666666666666) 0.0)
    -1.0))))

C

double code(double x, double y) {
	return x * (-1.0 / fma(y, fma(y, fma(y, fma(y, 0.008333333333333333, 0.0), -0.16666666666666666), 0.0), -1.0));
}

Julia

function code(x, y)
	return Float64(x * Float64(-1.0 / fma(y, fma(y, fma(y, fma(y, 0.008333333333333333, 0.0), -0.16666666666666666), 0.0), -1.0)))
end

Wolfram

code[x_, y_] := N[(x * N[(-1.0 / N[(y * N[(y * N[(y * N[(y * 0.008333333333333333 + 0.0), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 0.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. *-rgt-identity N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

5. Simplified 54.1%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y, 0\right), \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 1\right)} \]

7. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, y, 0\right), \mathsf{fma}\left(y, y, 0\right), 0\right), \mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 0\right), -1\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 0\right), -1\right)} \cdot x\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1} \cdot \left(\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \frac{1}{120}, 0\right), \frac{-1}{6}\right), 0\right), -1\right)} \cdot x\right) \]
9. Step-by-step derivation

10. Simplified 64.0%

    \[\leadsto \color{blue}{-1} \cdot \left(\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 0\right), -1\right)} \cdot x\right) \]

11. Final simplification 64.0%

    \[\leadsto x \cdot \frac{-1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 0\right), -1\right)} \]
12. Add Preprocessing

Alternative 5: 56.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5)
   (fma
    y
    (* y (* x (fma (* y y) 0.008333333333333333 -0.16666666666666666)))
    x)
   (/ 1.0 (/ (/ y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5) {
		tmp = fma(y, (y * (x * fma((y * y), 0.008333333333333333, -0.16666666666666666))), x);
	} else {
		tmp = 1.0 / ((y / x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5)
		tmp = fma(y, Float64(y * Float64(x * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666))), x);
	else
		tmp = Float64(1.0 / Float64(Float64(y / x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5], N[(y * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(1.0 / N[(N[(y / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5

   1. Initial program 99.9%

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

      1. associate-*r* N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. sin-lowering-sin.f64 89.7

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

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   9. Simplified 70.0%

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

   if 4.5 < y

   1. Initial program 99.7%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.2

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

   7. Simplified 4.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. clear-num N/A

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

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

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

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

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

      9. /-lowering-/.f64 23.6

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

   9. Applied egg-rr 23.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5)
   (fma
    y
    (* y (* x (fma (* y y) 0.008333333333333333 -0.16666666666666666)))
    x)
   (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5) {
		tmp = fma(y, (y * (x * fma((y * y), 0.008333333333333333, -0.16666666666666666))), x);
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5)
		tmp = fma(y, Float64(y * Float64(x * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666))), x);
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5], N[(y * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5

   1. Initial program 99.9%

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

      1. associate-*r* N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. sin-lowering-sin.f64 89.7

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

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   9. Simplified 70.0%

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

   if 4.5 < y

   1. Initial program 99.7%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.2

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

   7. Simplified 4.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

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

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

      7. /-lowering-/.f64 23.6

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

   9. Applied egg-rr 23.6%

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

Alternative 7: 56.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5)
   (*
    x
    (fma (* y y) (fma y (* y 0.008333333333333333) -0.16666666666666666) 1.0))
   (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5) {
		tmp = x * fma((y * y), fma(y, (y * 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5)
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5

   1. Initial program 99.9%

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

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

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

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

   7. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.008333333333333333, 0\right), -0.16666666666666666\right), 1\right) \cdot x} \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 70.0

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

   10. Simplified 70.0%

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

   if 4.5 < y

   1. Initial program 99.7%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.2

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

   7. Simplified 4.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

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

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

      7. /-lowering-/.f64 23.6

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

   9. Applied egg-rr 23.6%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot -0.16666666666666666\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3200.0) (fma (* x (* y -0.16666666666666666)) y x) (/ y (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3200.0) {
		tmp = fma((x * (y * -0.16666666666666666)), y, x);
	} else {
		tmp = y / (y / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3200.0)
		tmp = fma(Float64(x * Float64(y * -0.16666666666666666)), y, x);
	else
		tmp = Float64(y / Float64(y / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3200.0], N[(N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 69.8

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

   5. Simplified 69.8%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

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

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

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

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

      7. *-lowering-*.f64 69.8

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

   7. Applied egg-rr 69.8%

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

   if 3200 < y

   1. Initial program 99.7%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 4.2

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

   7. Simplified 4.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

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

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

      7. /-lowering-/.f64 23.6

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

   9. Applied egg-rr 23.6%

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

Alternative 9: 56.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot -0.16666666666666666\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3200.0) (fma (* x (* y -0.16666666666666666)) y x) (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3200.0) {
		tmp = fma((x * (y * -0.16666666666666666)), y, x);
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3200.0)
		tmp = fma(Float64(x * Float64(y * -0.16666666666666666)), y, x);
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3200.0], N[(N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 69.8

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

   5. Simplified 69.8%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

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

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

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

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

      7. *-lowering-*.f64 69.8

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

   7. Applied egg-rr 69.8%

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

   if 3200 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   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 23.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot -0.16666666666666666\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3200.0) (* x (fma y (* y -0.16666666666666666) 1.0)) (* y (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3200.0) {
		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 <= 3200.0)
		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, 3200.0], 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 < 3200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-in N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 69.8

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

   5. Simplified 69.8%

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

   if 3200 < y

   1. Initial program 99.7%

      \[x \cdot \frac{\sin y}{y} \]
   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 23.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3200:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.7% 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 54.7%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))