Data.Number.Erf:$dmerfcx from erf-2.0.0.0

Percentage Accurate: 100.0% → 100.0%

Time: 35.9s

Alternatives: 15

Speedup: 1.0×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot 1\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

   5. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\frac{y}{2} + y \cdot \frac{1}{2}\right)\right)\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\frac{y}{2} + \frac{y}{2}\right)\right)\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

   9. +-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{0}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

   10. +-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{0}{0}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{y \cdot 0}{0}\right)\right)\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot 1\right) \cdot 0}{0}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

   16. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right)\right) \]

   17. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + y \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right)\right) \]

   18. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot 0}{0}\right)\right)\right) \]

   19. +-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \left(\frac{y}{2} - \frac{y}{2}\right)}{0}\right)\right)\right) \]

   20. difference-of-squares N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{0}\right)\right)\right) \]

   21. +-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

   22. flip-+ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{y}{2} + \frac{y}{2}\right)\right)\right) \]

   23. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(2 \cdot \frac{y}{2}\right)\right)\right) \]

4. Applied egg-rr 73.4%

   \[\leadsto x \cdot e^{\color{blue}{y}} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + y \cdot \left(y \cdot \left(1 + y \cdot \left(\left(y \cdot \left(4 + \left(y \cdot y\right) \cdot 1.3333333333333333\right)\right) \cdot \left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    y
    (*
     y
     (+
      1.0
      (*
       y
       (*
        (* y (+ 4.0 (* (* y y) 1.3333333333333333)))
        (+
         0.125
         (* (* y y) (* (* y (* y (* y y))) 0.004629629629629629)))))))))))

C

double code(double x, double y) {
	return x * (1.0 + (y * (y * (1.0 + (y * ((y * (4.0 + ((y * y) * 1.3333333333333333))) * (0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629)))))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * (y * (1.0d0 + (y * ((y * (4.0d0 + ((y * y) * 1.3333333333333333d0))) * (0.125d0 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629d0)))))))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * (y * (1.0 + (y * ((y * (4.0 + ((y * y) * 1.3333333333333333))) * (0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629)))))))));
}

Python

def code(x, y):
	return x * (1.0 + (y * (y * (1.0 + (y * ((y * (4.0 + ((y * y) * 1.3333333333333333))) * (0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629)))))))))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * Float64(y * Float64(1.0 + Float64(y * Float64(Float64(y * Float64(4.0 + Float64(Float64(y * y) * 1.3333333333333333))) * Float64(0.125 + Float64(Float64(y * y) * Float64(Float64(y * Float64(y * Float64(y * y))) * 0.004629629629629629))))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * (y * (1.0 + (y * ((y * (4.0 + ((y * y) * 1.3333333333333333))) * (0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629)))))))));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * N[(y * N[(1.0 + N[(y * N[(N[(y * N[(4.0 + N[(N[(y * y), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 95.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right), \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

   10. *-lowering-*.f64 95.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

7. Applied egg-rr 95.4%

   \[\leadsto x \cdot \left(1 + \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right) \cdot y}\right) \]

9. Applied egg-rr 59.9%

   \[\leadsto x \cdot \left(1 + \left(y \cdot \left(1 + y \cdot \color{blue}{\left(\frac{y}{y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) + -0.08333333333333333\right)\right) + 0.25} \cdot \left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right)\right)}\right)\right) \cdot y\right) \]

10. Taylor expanded in y around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{\left(y \cdot \left(4 + \frac{4}{3} \cdot {y}^{2}\right)\right)}, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(4 + \frac{4}{3} \cdot {y}^{2}\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(4, \left(\frac{4}{3} \cdot {y}^{2}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(4, \left({y}^{2} \cdot \frac{4}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{4}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{4}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

    6. *-lowering-*.f64 97.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{4}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

12. Simplified 97.4%

    \[\leadsto x \cdot \left(1 + \left(y \cdot \left(1 + y \cdot \left(\color{blue}{\left(y \cdot \left(4 + \left(y \cdot y\right) \cdot 1.3333333333333333\right)\right)} \cdot \left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right)\right)\right)\right) \cdot y\right) \]

13. Final simplification 97.4%

    \[\leadsto x \cdot \left(1 + y \cdot \left(y \cdot \left(1 + y \cdot \left(\left(y \cdot \left(4 + \left(y \cdot y\right) \cdot 1.3333333333333333\right)\right) \cdot \left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 4: 96.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + y \cdot \left(y \cdot \left(1 + y \cdot \left(\left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right) \cdot \left(y \cdot 4\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    y
    (*
     y
     (+
      1.0
      (*
       y
       (*
        (+ 0.125 (* (* y y) (* (* y (* y (* y y))) 0.004629629629629629)))
        (* y 4.0)))))))))

C

double code(double x, double y) {
	return x * (1.0 + (y * (y * (1.0 + (y * ((0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629))) * (y * 4.0)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * (y * (1.0d0 + (y * ((0.125d0 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629d0))) * (y * 4.0d0)))))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * (y * (1.0 + (y * ((0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629))) * (y * 4.0)))))));
}

Python

def code(x, y):
	return x * (1.0 + (y * (y * (1.0 + (y * ((0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629))) * (y * 4.0)))))))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * Float64(y * Float64(1.0 + Float64(y * Float64(Float64(0.125 + Float64(Float64(y * y) * Float64(Float64(y * Float64(y * Float64(y * y))) * 0.004629629629629629))) * Float64(y * 4.0))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * (y * (1.0 + (y * ((0.125 + ((y * y) * ((y * (y * (y * y))) * 0.004629629629629629))) * (y * 4.0)))))));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * N[(y * N[(1.0 + N[(y * N[(N[(0.125 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 95.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right), \color{blue}{y}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{2} + \left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

   10. *-lowering-*.f64 95.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

7. Applied egg-rr 95.4%

   \[\leadsto x \cdot \left(1 + \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right) \cdot y}\right) \]

9. Applied egg-rr 59.9%

   \[\leadsto x \cdot \left(1 + \left(y \cdot \left(1 + y \cdot \color{blue}{\left(\frac{y}{y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) + -0.08333333333333333\right)\right) + 0.25} \cdot \left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right)\right)}\right)\right) \cdot y\right) \]

10. Taylor expanded in y around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{\left(4 \cdot y\right)}, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot 4\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

    2. *-lowering-*.f64 96.5%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \frac{1}{216}\right)\right)\right)\right)\right)\right)\right), y\right)\right)\right) \]

12. Simplified 96.5%

    \[\leadsto x \cdot \left(1 + \left(y \cdot \left(1 + y \cdot \left(\color{blue}{\left(y \cdot 4\right)} \cdot \left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right)\right)\right)\right) \cdot y\right) \]

13. Final simplification 96.5%

    \[\leadsto x \cdot \left(1 + y \cdot \left(y \cdot \left(1 + y \cdot \left(\left(0.125 + \left(y \cdot y\right) \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 0.004629629629629629\right)\right) \cdot \left(y \cdot 4\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 5: 93.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot y \leq 4:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.16666666666666666 \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))))
   (if (<= (* y y) 4.0)
     (+ x (* x (* y y)))
     (* x (* 0.16666666666666666 (* t_0 t_0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if ((y * y) <= 4.0) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (0.16666666666666666 * (t_0 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * y)
    if ((y * y) <= 4.0d0) then
        tmp = x + (x * (y * y))
    else
        tmp = x * (0.16666666666666666d0 * (t_0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if ((y * y) <= 4.0) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (0.16666666666666666 * (t_0 * t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * y)
	tmp = 0
	if (y * y) <= 4.0:
		tmp = x + (x * (y * y))
	else:
		tmp = x * (0.16666666666666666 * (t_0 * t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (Float64(y * y) <= 4.0)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(x * Float64(0.16666666666666666 * Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * y);
	tmp = 0.0;
	if ((y * y) <= 4.0)
		tmp = x + (x * (y * y));
	else
		tmp = x * (0.16666666666666666 * (t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * y), $MachinePrecision], 4.0], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 98.3%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

         \[\leadsto \left(y \cdot y\right) \cdot x + x \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \color{blue}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), x\right) \]

      6. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), x\right) \]

   7. Applied egg-rr 98.3%

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

   if 4 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 91.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(y \cdot \frac{1}{6}\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(y \cdot \frac{1}{6}\right), \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 91.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), y\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 91.6%

      \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \color{blue}{\left(y \cdot 0.16666666666666666\right) \cdot y}\right)\right)\right)\right) \]

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{6}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{6}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right)\right) \]

      7. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{3} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left({y}^{3}\right)}\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left({\color{blue}{y}}^{3}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left({y}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left({\color{blue}{y}}^{3}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left({y}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{3}\right)\right)\right)\right) \]

      14. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 91.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   10. Simplified 91.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.16666666666666666 \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (+
   1.0
   (* (* y y) (+ 1.0 (* y (* y (+ 0.5 (* y (* y 0.16666666666666666))))))))))

C

double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + (y * (y * 0.16666666666666666))))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((y * y) * (1.0d0 + (y * (y * (0.5d0 + (y * (y * 0.16666666666666666d0))))))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + (y * (y * 0.16666666666666666))))))));
}

Python

def code(x, y):
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + (y * (y * 0.16666666666666666))))))))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(Float64(y * y) * Float64(1.0 + Float64(y * Float64(y * Float64(0.5 + Float64(y * Float64(y * 0.16666666666666666)))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + (y * (y * 0.16666666666666666))))))));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(0.5 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 95.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(y \cdot \frac{1}{6}\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(y \cdot \frac{1}{6}\right), \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 95.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), y\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 95.4%

   \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \color{blue}{\left(y \cdot 0.16666666666666666\right) \cdot y}\right)\right)\right)\right) \]

8. Final simplification 95.4%

   \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 7: 90.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.4:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.4)
   (+ x (* x (* y y)))
   (* x (* y (* y (+ 1.0 (* (* y y) 0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (y * (y * (1.0 + ((y * y) * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 0.4d0) then
        tmp = x + (x * (y * y))
    else
        tmp = x * (y * (y * (1.0d0 + ((y * y) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (y * (y * (1.0 + ((y * y) * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.4:
		tmp = x + (x * (y * y))
	else:
		tmp = x * (y * (y * (1.0 + ((y * y) * 0.5))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.4)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(x * Float64(y * Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.4)
		tmp = x + (x * (y * y));
	else
		tmp = x * (y * (y * (1.0 + ((y * y) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.4], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.40000000000000002

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 99.5%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

         \[\leadsto \left(y \cdot y\right) \cdot x + x \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \color{blue}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), x\right) \]

      6. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), x\right) \]

   7. Applied egg-rr 99.5%

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

   if 0.40000000000000002 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(y \cdot \frac{1}{6}\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(y \cdot \frac{1}{6}\right), \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{6}\right), y\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 90.4%

      \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \color{blue}{\left(y \cdot 0.16666666666666666\right) \cdot y}\right)\right)\right)\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{6} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}} + \frac{1}{{y}^{4}}\right)\right)\right)}\right) \]

   10. Simplified 90.4%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 87.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

   13. Simplified 87.8%

       \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.5\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.4:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* (* y y) (+ 1.0 (* y (* y (* (* y y) 0.16666666666666666))))))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((y * y) * (1.0d0 + (y * (y * ((y * y) * 0.16666666666666666d0))))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * ((y * y) * 0.16666666666666666))))));
}

Python

def code(x, y):
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * ((y * y) * 0.16666666666666666))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * ((y * y) * 0.16666666666666666))))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 95.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right)\right)\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 95.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 95.3%

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

9. Final simplification 95.3%

   \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 90.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4.0) (+ x (* x (* y y))) (* x (* y (* (* y (* y y)) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4.0) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (y * ((y * (y * y)) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 4.0d0) then
        tmp = x + (x * (y * y))
    else
        tmp = x * (y * ((y * (y * y)) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4.0) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (y * ((y * (y * y)) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4.0:
		tmp = x + (x * (y * y))
	else:
		tmp = x * (y * ((y * (y * y)) * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4.0)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(x * Float64(y * Float64(Float64(y * Float64(y * y)) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 4.0)
		tmp = x + (x * (y * y));
	else
		tmp = x * (y * ((y * (y * y)) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4.0], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 98.3%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

         \[\leadsto \left(y \cdot y\right) \cdot x + x \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \color{blue}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), x\right) \]

      6. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), x\right) \]

   7. Applied egg-rr 98.3%

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

   if 4 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 89.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + \left(y \cdot y\right) \cdot 0.5\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot {y}^{2}\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

      11. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{3}\right)}\right)\right)\right) \]

      13. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 89.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   8. Simplified 89.0%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + \left(y \cdot y\right) \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* (* y y) (+ 1.0 (* (* y y) 0.5))))))

C

double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + ((y * y) * 0.5))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((y * y) * (1.0d0 + ((y * y) * 0.5d0))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + ((y * y) * 0.5))));
}

Python

def code(x, y):
	return x * (1.0 + ((y * y) * (1.0 + ((y * y) * 0.5))))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(Float64(y * y) * Float64(1.0 + Float64(Float64(y * y) * 0.5)))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + ((y * y) * (1.0 + ((y * y) * 0.5))));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-in N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 94.2%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

5. Simplified 94.2%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + \left(y \cdot y\right) \cdot 0.5\right)\right)} \]
6. Add Preprocessing

Alternative 11: 81.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.4) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 0.4d0) then
        tmp = x
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.4:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.4)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.4)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.4], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.40000000000000002

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 99.1%

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

   if 0.40000000000000002 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 64.1%

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

   6. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 64.1%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.8% accurate, 13.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) x (* x y)))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x
	else:
		tmp = x * y
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 69.4%

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

   if 1 < y

   1. Initial program 100.0%

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

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot 1\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\frac{y}{2} + y \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\frac{y}{2} + \frac{y}{2}\right)\right)\right)\right) \]

      8. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{0}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{0}{0}\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{y \cdot 0}{0}\right)\right)\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot 1\right) \cdot 0}{0}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

      16. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right)\right) \]

      17. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + y \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right)\right) \]

      18. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot 0}{0}\right)\right)\right) \]

      19. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \left(\frac{y}{2} - \frac{y}{2}\right)}{0}\right)\right)\right) \]

      20. difference-of-squares N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{0}\right)\right)\right) \]

      21. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

      22. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{y}{2} + \frac{y}{2}\right)\right)\right) \]

      23. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(2 \cdot \frac{y}{2}\right)\right)\right) \]

   4. Applied egg-rr 96.6%

      \[\leadsto x \cdot e^{\color{blue}{y}} \]

   5. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-out N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 30.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right) \]

   7. Simplified 30.2%

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

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 30.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 30.2%

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

Alternative 13: 81.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ x + x \cdot \left(y \cdot y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-rgt-identity N/A

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

   2. distribute-lft-in N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

   7. *-lowering-*.f64 83.5%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

5. Simplified 83.5%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

      \[\leadsto \left(y \cdot y\right) \cdot x + x \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \color{blue}{x}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), x\right) \]

   6. *-lowering-*.f64 83.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), x\right) \]

7. Applied egg-rr 83.5%

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

8. Final simplification 83.5%

   \[\leadsto x + x \cdot \left(y \cdot y\right) \]
9. Add Preprocessing

Alternative 14: 81.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y \cdot y + 1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-rgt-identity N/A

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

   2. distribute-lft-in N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

   7. *-lowering-*.f64 83.5%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

5. Simplified 83.5%

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

Alternative 15: 51.9% accurate, 105.0× 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 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 56.1%

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

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot {\left(e^{y}\right)}^{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (pow (exp y) y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * math.pow(math.exp(y), y)

Julia

function code(x, y)
	return Float64(x * (exp(y) ^ y))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (pow (exp y) y)))

  (* x (exp (* y y))))