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

Percentage Accurate: 100.0% → 99.5%

Time: 35.8s

Alternatives: 18

Speedup: 1.0×

• 49.3% 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: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m \cdot y\_m}{\mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.08333333333333333, -0.5\right), 1\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 0.001)
   (fma
    (/
     (* y_m y_m)
     (fma y_m (* y_m (fma (* y_m y_m) 0.08333333333333333 -0.5)) 1.0))
    x
    x)
   (* x (exp y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = fma(((y_m * y_m) / fma(y_m, (y_m * fma((y_m * y_m), 0.08333333333333333, -0.5)), 1.0)), x, x);
	} else {
		tmp = x * exp(y_m);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 0.001)
		tmp = fma(Float64(Float64(y_m * y_m) / fma(y_m, Float64(y_m * fma(Float64(y_m * y_m), 0.08333333333333333, -0.5)), 1.0)), x, x);
	else
		tmp = Float64(x * exp(y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 0.001], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] / N[(y$95$m * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(x * N[Exp[y$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-3

   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 99.9%

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

      1. flip3-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}}, 1\right) \]

      2. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{\frac{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}}}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{\frac{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}}}, 1\right) \]

      4. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{\frac{1}{\frac{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}}}}, 1\right) \]

      5. flip3-+ N/A

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

   7. Applied egg-rr 99.9%

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), 1\right)}}}, 1\right) \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{12} \cdot {y}^{2} - \frac{1}{2}\right) + 1}}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{12} \cdot {y}^{2} - \frac{1}{2}, 1\right)}}, 1\right) \]

      3. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{12} \cdot {y}^{2} - \frac{1}{2}, 1\right)}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{12} \cdot {y}^{2} - \frac{1}{2}, 1\right)}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{12} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{12}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \frac{1}{12} + \color{blue}{\frac{-1}{2}}, 1\right)}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{12}, \frac{-1}{2}\right)}, 1\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{12}, \frac{-1}{2}\right), 1\right)}, 1\right) \]

      10. *-lowering-*.f64 100.0

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.08333333333333333, -0.5\right), 1\right)}, 1\right) \]

   10. Simplified 100.0%

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.08333333333333333, -0.5\right), 1\right)}}, 1\right) \]
   11. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right) + 1}\right) \cdot x + 1 \cdot x} \]

       2. *-lft-identity N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot \frac{1}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right) + 1}\right) \cdot x + \color{blue}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right) + 1}, x, x\right)} \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y \cdot y}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right) + 1}}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y \cdot y}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right) + 1}}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y \cdot y}}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right) + 1}, x, x\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{\color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right)\right)} + 1}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right), 1\right)}}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{12} + \frac{-1}{2}\right)}, 1\right)}, x, x\right) \]

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

           \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{12}, \frac{-1}{2}\right)}, 1\right)}, x, x\right) \]

       11. *-lowering-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.08333333333333333, -0.5\right), 1\right)}, x, x\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y \cdot y}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.08333333333333333, -0.5\right), 1\right)}, x, x\right)} \]

   if 1e-3 < (*.f64 y 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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-out N/A

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. flip-+ N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      11. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out N/A

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

      17. div-inv N/A

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

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      19. +-inverses N/A

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

      20. difference-of-squares N/A

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

      21. +-inverses N/A

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

      22. flip-+ N/A

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

      23. count-2 N/A

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

      24. associate-*r/ N/A

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

   4. Applied egg-rr 50.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot e^{y\_m \cdot y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* x (exp (* y_m y_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[Exp[N[(y$95$m * y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 3: 95.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m \cdot y\_m, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right), 1\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (*
  x
  (fma
   (* y_m y_m)
   (fma
    y_m
    (*
     y_m
     (fma
      (* y_m y_m)
      (fma (* y_m y_m) 0.041666666666666664 0.16666666666666666)
      0.5))
    1.0)
   1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * fma((y_m * y_m), fma(y_m, (y_m * fma((y_m * y_m), fma((y_m * y_m), 0.041666666666666664, 0.16666666666666666), 0.5)), 1.0), 1.0);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * fma(Float64(y_m * y_m), fma(y_m, Float64(y_m * fma(Float64(y_m * y_m), fma(Float64(y_m * y_m), 0.041666666666666664, 0.16666666666666666), 0.5)), 1.0), 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $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

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

5. Simplified 94.7%

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

   1. flip3-+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}}, 1\right) \]

   2. clear-num N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{\frac{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}}}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{\frac{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}}}, 1\right) \]

   4. clear-num N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{\frac{1}{\frac{{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot 1\right)}}}}, 1\right) \]

   5. flip3-+ N/A

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

7. Applied egg-rr 94.7%

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), 1\right)}}}, 1\right) \]

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{12} \cdot {y}^{2} - \frac{1}{2}\right) + 1}}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{12} \cdot {y}^{2} - \frac{1}{2}, 1\right)}}, 1\right) \]

   3. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{12} \cdot {y}^{2} - \frac{1}{2}, 1\right)}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{12} \cdot {y}^{2} - \frac{1}{2}, 1\right)}, 1\right) \]

   5. sub-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{12} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}, 1\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{12}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}, 1\right) \]

   7. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \frac{1}{12} + \color{blue}{\frac{-1}{2}}, 1\right)}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{12}, \frac{-1}{2}\right)}, 1\right)}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{12}, \frac{-1}{2}\right), 1\right)}, 1\right) \]

   10. *-lowering-*.f64 54.8

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.08333333333333333, -0.5\right), 1\right)}, 1\right) \]

10. Simplified 54.8%

    \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.08333333333333333, -0.5\right), 1\right)}}, 1\right) \]

11. Taylor expanded in y around 0

    \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right)\right)}, 1\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right)\right) + 1}, 1\right) \]

    2. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right)\right) + 1, 1\right) \]

    3. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right)\right)\right)} + 1, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right)\right), 1\right)}, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right)\right)}, 1\right), 1\right) \]

    6. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot {y}^{2}\right) + \frac{1}{2}\right)}, 1\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{24} \cdot {y}^{2}, \frac{1}{2}\right)}, 1\right), 1\right) \]

    8. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{24} \cdot {y}^{2}, \frac{1}{2}\right), 1\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{24} \cdot {y}^{2}, \frac{1}{2}\right), 1\right), 1\right) \]

    10. +-commutative N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{24} \cdot {y}^{2} + \frac{1}{6}}, \frac{1}{2}\right), 1\right), 1\right) \]

    11. *-commutative N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \]

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

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24}, \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right), 1\right) \]

    13. unpow2 N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24}, \frac{1}{6}\right), \frac{1}{2}\right), 1\right), 1\right) \]

    14. *-lowering-*.f64 97.0

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right), 1\right) \]

13. Simplified 97.0%

    \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}, 1\right) \]
14. Add Preprocessing

Alternative 4: 93.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y\_m, x \cdot \left(y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.5, 1\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y\_m \cdot y\_m\right) \cdot \left(\left(y\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.16666666666666666, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 0.001)
   (fma y_m (* x (* y_m (fma (* y_m y_m) 0.5 1.0))) x)
   (*
    x
    (*
     (* y_m y_m)
     (* (* y_m y_m) (fma y_m (* y_m 0.16666666666666666) 0.5))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = fma(y_m, (x * (y_m * fma((y_m * y_m), 0.5, 1.0))), x);
	} else {
		tmp = x * ((y_m * y_m) * ((y_m * y_m) * fma(y_m, (y_m * 0.16666666666666666), 0.5)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 0.001)
		tmp = fma(y_m, Float64(x * Float64(y_m * fma(Float64(y_m * y_m), 0.5, 1.0))), x);
	else
		tmp = Float64(x * Float64(Float64(y_m * y_m) * Float64(Float64(y_m * y_m) * fma(y_m, Float64(y_m * 0.16666666666666666), 0.5))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 0.001], N[(y$95$m * N[(x * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-3

   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. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

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

      11. +-commutative N/A

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

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

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

      10. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   if 1e-3 < (*.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 88.7%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 88.7%

      \[\leadsto \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.5, 1\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y\_m, x \cdot \left(y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.5, 1\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.16666666666666666 \cdot \left(\left(y\_m \cdot y\_m\right) \cdot \left(y\_m \cdot \left(y\_m \cdot \left(y\_m \cdot y\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 0.001)
   (fma y_m (* x (* y_m (fma (* y_m y_m) 0.5 1.0))) x)
   (* x (* 0.16666666666666666 (* (* y_m y_m) (* y_m (* y_m (* y_m y_m))))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = fma(y_m, (x * (y_m * fma((y_m * y_m), 0.5, 1.0))), x);
	} else {
		tmp = x * (0.16666666666666666 * ((y_m * y_m) * (y_m * (y_m * (y_m * y_m)))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 0.001)
		tmp = fma(y_m, Float64(x * Float64(y_m * fma(Float64(y_m * y_m), 0.5, 1.0))), x);
	else
		tmp = Float64(x * Float64(0.16666666666666666 * Float64(Float64(y_m * y_m) * Float64(y_m * Float64(y_m * Float64(y_m * y_m))))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 0.001], N[(y$95$m * N[(x * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-3

   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. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

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

      11. +-commutative N/A

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

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

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

   5. Simplified 99.8%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

      10. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   if 1e-3 < (*.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 88.7%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. cube-prod N/A

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

      7. unpow2 N/A

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

      8. unpow3 N/A

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

      9. pow-sqr N/A

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

      10. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{1}{6} \cdot \left({y}^{\color{blue}{4}} \cdot {y}^{2}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{4}\right) \cdot {y}^{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{4} \cdot \frac{1}{6}\right)} \cdot {y}^{2}\right) \]

      13. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*r* N/A

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

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

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

      18. associate-*r* N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

          \[\leadsto x \cdot \left({y}^{\color{blue}{4}} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left({y}^{4} \cdot \frac{1}{6}\right) \cdot {y}^{2}\right)} \]

   8. Simplified 88.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.5, 1\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \mathsf{fma}\left(\left(y\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 0.5\right), 1\right), x, x\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma
  (*
   (* y_m y_m)
   (fma (* y_m y_m) (fma (* y_m y_m) 0.16666666666666666 0.5) 1.0))
  x
  x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma(((y_m * y_m) * fma((y_m * y_m), fma((y_m * y_m), 0.16666666666666666, 0.5), 1.0)), x, x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(Float64(Float64(y_m * y_m) * fma(Float64(y_m * y_m), fma(Float64(y_m * y_m), 0.16666666666666666, 0.5), 1.0)), x, x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x + x), $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 94.7%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-lowering-*.f64 94.8

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

7. Applied egg-rr 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), 1\right), x, x\right)} \]
8. Add Preprocessing

Alternative 7: 93.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (*
  x
  (fma
   (* y_m y_m)
   (fma y_m (* y_m (fma y_m (* y_m 0.16666666666666666) 0.5)) 1.0)
   1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * fma((y_m * y_m), fma(y_m, (y_m * fma(y_m, (y_m * 0.16666666666666666), 0.5)), 1.0), 1.0);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * fma(Float64(y_m * y_m), fma(y_m, Float64(y_m * fma(y_m, Float64(y_m * 0.16666666666666666), 0.5)), 1.0), 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $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

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

5. Simplified 94.7%

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

Alternative 8: 93.7% accurate, 2.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma
  (* (* y_m y_m) (fma (* y_m y_m) (* (* y_m y_m) 0.16666666666666666) 1.0))
  x
  x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma(((y_m * y_m) * fma((y_m * y_m), ((y_m * y_m) * 0.16666666666666666), 1.0)), x, x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(Float64(Float64(y_m * y_m) * fma(Float64(y_m * y_m), Float64(Float64(y_m * y_m) * 0.16666666666666666), 1.0)), x, x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x + x), $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 94.7%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-lowering-*.f64 94.8

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

7. Applied egg-rr 94.8%

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

8. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 94.6

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

10. Simplified 94.6%

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

Alternative 9: 91.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y\_m, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot \left(0.5 \cdot \left(y\_m \cdot \left(y\_m \cdot y\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 0.001)
   (fma (* x y_m) y_m x)
   (* x (* y_m (* 0.5 (* y_m (* y_m y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = fma((x * y_m), y_m, x);
	} else {
		tmp = x * (y_m * (0.5 * (y_m * (y_m * y_m))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 0.001)
		tmp = fma(Float64(x * y_m), y_m, x);
	else
		tmp = Float64(x * Float64(y_m * Float64(0.5 * Float64(y_m * Float64(y_m * y_m)))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 0.001], N[(N[(x * y$95$m), $MachinePrecision] * y$95$m + x), $MachinePrecision], N[(x * N[(y$95$m * N[(0.5 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-3

   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

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

      2. distribute-lft-in N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 99.6

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

   5. Simplified 99.6%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 1e-3 < (*.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. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

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

      11. +-commutative N/A

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

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

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

   5. Simplified 84.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({y}^{4} \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

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

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

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

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

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 84.6

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

   8. Simplified 84.6%

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

Alternative 10: 93.4% accurate, 3.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (*
  x
  (fma (* y_m y_m) (* y_m (* 0.16666666666666666 (* y_m (* y_m y_m)))) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * fma((y_m * y_m), (y_m * (0.16666666666666666 * (y_m * (y_m * y_m)))), 1.0);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * fma(Float64(y_m * y_m), Float64(y_m * Float64(0.16666666666666666 * Float64(y_m * Float64(y_m * y_m)))), 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

5. Simplified 94.7%

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

6. Taylor expanded in y around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 94.0

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

8. Simplified 94.0%

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

Alternative 11: 91.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m, y\_m \cdot 0.5, 1\right), 1\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* x (fma (* y_m y_m) (fma y_m (* y_m 0.5) 1.0) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * fma((y_m * y_m), fma(y_m, (y_m * 0.5), 1.0), 1.0);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * fma(Float64(y_m * y_m), fma(y_m, Float64(y_m * 0.5), 1.0), 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * 0.5), $MachinePrecision] + 1.0), $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

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

   1. distribute-rgt-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-in N/A

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

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

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

   11. +-commutative N/A

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

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

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

5. Simplified 92.8%

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

Alternative 12: 90.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \left(y\_m \cdot y\_m\right) \cdot 0.5, 1\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* x (fma (* y_m y_m) (* (* y_m y_m) 0.5) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * fma((y_m * y_m), ((y_m * y_m) * 0.5), 1.0);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * fma(Float64(y_m * y_m), Float64(Float64(y_m * y_m) * 0.5), 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.5), $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

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

   1. distribute-rgt-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-in N/A

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

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

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

   11. +-commutative N/A

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

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

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

5. Simplified 92.8%

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

6. Taylor expanded in y around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 92.2

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

8. Simplified 92.2%

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

9. Final simplification 92.2%

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot 0.5, 1\right) \]
10. Add Preprocessing

Alternative 13: 87.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \mathsf{fma}\left(x, y\_m \cdot \mathsf{fma}\left(y\_m, \mathsf{fma}\left(y\_m, 0.16666666666666666, 0.5\right), 1\right), x\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma x (* y_m (fma y_m (fma y_m 0.16666666666666666 0.5) 1.0)) x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma(x, (y_m * fma(y_m, fma(y_m, 0.16666666666666666, 0.5), 1.0)), x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(x, Float64(y_m * fma(y_m, fma(y_m, 0.16666666666666666, 0.5), 1.0)), x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + x), $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

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

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

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-out N/A

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

   17. div-inv N/A

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

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

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

   20. difference-of-squares N/A

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

   21. +-inverses N/A

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

   22. flip-+ N/A

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

   23. count-2 N/A

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

   24. associate-*r/ N/A

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

4. Applied egg-rr 75.2%

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

5. Taylor expanded in y around 0

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

7. Simplified 69.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), x\right)} \]
8. Add Preprocessing

Alternative 14: 81.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 0.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y\_m \cdot y\_m\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 0.001) x (* x (* y_m y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = x;
	} else {
		tmp = x * (y_m * y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m * y_m) <= 0.001d0) then
        tmp = x
    else
        tmp = x * (y_m * y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = x;
	} else {
		tmp = x * (y_m * y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m * y_m) <= 0.001:
		tmp = x
	else:
		tmp = x * (y_m * y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 0.001)
		tmp = x;
	else
		tmp = Float64(x * Float64(y_m * y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m * y_m) <= 0.001)
		tmp = x;
	else
		tmp = x * (y_m * y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 0.001], x, N[(x * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-3

   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 98.6%

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

   if 1e-3 < (*.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

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

      2. distribute-lft-in N/A

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 66.1

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

   5. Simplified 66.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 66.1

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

   8. Simplified 66.1%

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

Alternative 15: 64.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 0.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= (* y_m y_m) 0.001) x (* x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = x;
	} else {
		tmp = x * y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m * y_m) <= 0.001d0) then
        tmp = x
    else
        tmp = x * y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 0.001) {
		tmp = x;
	} else {
		tmp = x * y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m * y_m) <= 0.001:
		tmp = x
	else:
		tmp = x * y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 0.001)
		tmp = x;
	else
		tmp = Float64(x * y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m * y_m) <= 0.001)
		tmp = x;
	else
		tmp = x * y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 0.001], x, N[(x * y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-3

   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 98.6%

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

   if 1e-3 < (*.f64 y 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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-out N/A

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. flip-+ N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      11. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out N/A

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

      17. div-inv N/A

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

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      19. +-inverses N/A

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

      20. difference-of-squares N/A

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

      21. +-inverses N/A

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

      22. flip-+ N/A

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

      23. count-2 N/A

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

      24. associate-*r/ N/A

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 13.7

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

   7. Simplified 13.7%

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 13.7

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

   9. Applied egg-rr 13.7%

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

   10. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 13.7

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

   12. Simplified 13.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \mathsf{fma}\left(y\_m, y\_m, 1\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* x (fma y_m y_m 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * fma(y_m, y_m, 1.0);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * fma(y_m, y_m, 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(y$95$m * y$95$m + 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

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

   2. distribute-lft-in N/A

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

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

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. accelerator-lowering-fma.f64 84.2

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

5. Simplified 84.2%

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

Alternative 17: 63.8% accurate, 15.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \mathsf{fma}\left(x, y\_m, x\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (fma x y_m x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma(x, y_m, x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(x, y_m, x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * y$95$m + x), $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

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

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

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-out N/A

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

   17. div-inv N/A

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

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

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

   20. difference-of-squares N/A

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

   21. +-inverses N/A

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

   22. flip-+ N/A

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

   23. count-2 N/A

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

   24. associate-*r/ N/A

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

4. Applied egg-rr 75.2%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 58.5

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

7. Simplified 58.5%

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

Alternative 18: 51.9% accurate, 111.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 x)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x;
}

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x

Julia

y_m = abs(y)
function code(x, y_m)
	return x
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := 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 54.9%

   \[\leadsto \color{blue}{x} \]
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 2024196 
(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))))