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

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 13

Speedup: 1.0×

• 49.4% 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: 73.8% 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 75.2%

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

Alternative 3: 95.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\\ t_1 := \left(y \cdot y\right) \cdot \left(-1 - t\_0\right)\\ \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(\left(y \cdot y\right) \cdot \left(1 + t\_0\right)\right) \cdot t\_1\right)}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y (+ 0.5 (* y (* y 0.16666666666666666))))))
        (t_1 (* (* y y) (- -1.0 t_0))))
   (if (<= (* y y) 2e+92)
     (/ (* x (+ 1.0 (* (* (* y y) (+ 1.0 t_0)) t_1))) (+ 1.0 t_1))
     (+ x (* (* y (* y (* (* y y) 0.16666666666666666))) (* x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * (0.5 + (y * (y * 0.16666666666666666))));
	double t_1 = (y * y) * (-1.0 - t_0);
	double tmp;
	if ((y * y) <= 2e+92) {
		tmp = (x * (1.0 + (((y * y) * (1.0 + t_0)) * t_1))) / (1.0 + t_1);
	} else {
		tmp = x + ((y * (y * ((y * y) * 0.16666666666666666))) * (x * (y * y)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * (y * (0.5d0 + (y * (y * 0.16666666666666666d0))))
    t_1 = (y * y) * ((-1.0d0) - t_0)
    if ((y * y) <= 2d+92) then
        tmp = (x * (1.0d0 + (((y * y) * (1.0d0 + t_0)) * t_1))) / (1.0d0 + t_1)
    else
        tmp = x + ((y * (y * ((y * y) * 0.16666666666666666d0))) * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * (0.5 + (y * (y * 0.16666666666666666))));
	double t_1 = (y * y) * (-1.0 - t_0);
	double tmp;
	if ((y * y) <= 2e+92) {
		tmp = (x * (1.0 + (((y * y) * (1.0 + t_0)) * t_1))) / (1.0 + t_1);
	} else {
		tmp = x + ((y * (y * ((y * y) * 0.16666666666666666))) * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * (0.5 + (y * (y * 0.16666666666666666))))
	t_1 = (y * y) * (-1.0 - t_0)
	tmp = 0
	if (y * y) <= 2e+92:
		tmp = (x * (1.0 + (((y * y) * (1.0 + t_0)) * t_1))) / (1.0 + t_1)
	else:
		tmp = x + ((y * (y * ((y * y) * 0.16666666666666666))) * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * Float64(0.5 + Float64(y * Float64(y * 0.16666666666666666)))))
	t_1 = Float64(Float64(y * y) * Float64(-1.0 - t_0))
	tmp = 0.0
	if (Float64(y * y) <= 2e+92)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(Float64(Float64(y * y) * Float64(1.0 + t_0)) * t_1))) / Float64(1.0 + t_1));
	else
		tmp = Float64(x + Float64(Float64(y * Float64(y * Float64(Float64(y * y) * 0.16666666666666666))) * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * (0.5 + (y * (y * 0.16666666666666666))));
	t_1 = (y * y) * (-1.0 - t_0);
	tmp = 0.0;
	if ((y * y) <= 2e+92)
		tmp = (x * (1.0 + (((y * y) * (1.0 + t_0)) * t_1))) / (1.0 + t_1);
	else
		tmp = x + ((y * (y * ((y * y) * 0.16666666666666666))) * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(0.5 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * y), $MachinePrecision], 2e+92], N[(N[(x * N[(1.0 + N[(N[(N[(y * y), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.0000000000000001e92

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 93.3%

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

   if 2.0000000000000001e92 < (*.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 100.0%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)\right), x\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(-1 - y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)}{1 + \left(y \cdot y\right) \cdot \left(-1 - y \cdot \left(y \cdot \left(0.5 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (+
   1.0
   (* (* y y) (+ 1.0 (* y (* y (+ 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(Float64(y * 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[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $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 94.0%

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

Alternative 5: 90.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-5)
   (+ 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) <= 5e-5) {
		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) <= 5d-5) 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) <= 5e-5) {
		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) <= 5e-5:
		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) <= 5e-5)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(x * Float64(y * Float64(y * Float64(1.0 + Float64(y * Float64(y * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-5)
		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], 5e-5], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(1.0 + N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.00000000000000024e-5

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

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

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

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

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

      5. *-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 5.00000000000000024e-5 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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. unpow2 N/A

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

      18. associate-*l* N/A

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

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

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

      20. *-lowering-*.f64 83.0%

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

   5. Simplified 83.0%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. associate-*l* N/A

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

      16. distribute-rgt-in N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

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

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

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

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

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

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

      22. unpow2 N/A

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

      23. associate-*r* N/A

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

   8. Simplified 83.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

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 * (0.16666666666666666d0 * (y * (y * y)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(0.16666666666666666 * N[(y * N[(y * y), $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 94.0%

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

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. *-lowering-*.f64 N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 93.7%

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

8. Simplified 93.7%

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

Alternative 7: 90.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-5) (+ x (* x (* y y))) (* x (* y (* 0.5 (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-5) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (y * (0.5 * (y * (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) <= 5d-5) then
        tmp = x + (x * (y * y))
    else
        tmp = x * (y * (0.5d0 * (y * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-5) {
		tmp = x + (x * (y * y));
	} else {
		tmp = x * (y * (0.5 * (y * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-5:
		tmp = x + (x * (y * y))
	else:
		tmp = x * (y * (0.5 * (y * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-5)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-5)
		tmp = x + (x * (y * y));
	else
		tmp = x * (y * (0.5 * (y * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-5], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.00000000000000024e-5

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

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

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

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

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

      5. *-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 5.00000000000000024e-5 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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. unpow2 N/A

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

      18. associate-*l* N/A

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

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

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

      20. *-lowering-*.f64 83.0%

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

   5. Simplified 83.0%

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

   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. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 83.0%

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

   8. Simplified 83.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x + N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(y * y), $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 94.0%

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

6. Taylor expanded in y around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

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

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 94.0%

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

8. Simplified 94.0%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

10. Applied egg-rr 94.0%

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

11. Taylor expanded in y around inf

    \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right)\right), x\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

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

    2. pow-sqr N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. unpow2 N/A

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

    6. associate-*l* N/A

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

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

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

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

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

    9. *-commutative N/A

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

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

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

    11. unpow2 N/A

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

    12. *-lowering-*.f64 93.4%

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

13. Simplified 93.4%

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

14. Final simplification 93.4%

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

Alternative 9: 90.8% accurate, 7.0× speedup

Math

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

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(y * Float64(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[(y * N[(y * 0.5), $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

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
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. unpow2 N/A

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

   18. associate-*l* N/A

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

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

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

   20. *-lowering-*.f64 92.1%

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

5. Simplified 92.1%

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

Alternative 10: 81.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 5e-5) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-5) {
		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) <= 5d-5) 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) <= 5e-5) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-5:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-5)
		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) <= 5e-5)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-5], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.00000000000000024e-5

   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 98.8%

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

   if 5.00000000000000024e-5 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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 66.8%

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

   5. Simplified 66.8%

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

   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 66.8%

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

   8. Simplified 66.8%

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

Alternative 11: 81.9% 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 84.4%

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

5. Simplified 84.4%

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

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

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

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

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

   5. *-lowering-*.f64 84.4%

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

7. Applied egg-rr 84.4%

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

8. Final simplification 84.4%

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

Alternative 12: 81.9% 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 84.4%

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

5. Simplified 84.4%

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

Alternative 13: 51.7% 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 55.1%

   \[\leadsto \color{blue}{x} \]
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 2024145 
(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))))