Average Error: 5.6 → 0.1

Time: 6.9s

Precision: binary64

Cost: 708

\[x \cdot \left(1 + y \cdot y\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+183) (+ x (* (* y y) x)) (* y (* y x))))

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

↓

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+183) {
		tmp = x + ((y * y) * x);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 2d+183) then
        tmp = x + ((y * y) * x)
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

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

↓

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+183) {
		tmp = x + ((y * y) * x);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

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

↓

def code(x, y):
	tmp = 0
	if (y * y) <= 2e+183:
		tmp = x + ((y * y) * x)
	else:
		tmp = y * (y * x)
	return tmp

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

↓

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+183)
		tmp = Float64(x + Float64(Float64(y * y) * x));
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

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

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e+183)
		tmp = x + ((y * y) * x);
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

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

↓

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+183], N[(x + N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

x \cdot \left(1 + y \cdot y\right)

↓

\begin{array}{l}
\mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+183}:\\
\;\;\;\;x + \left(y \cdot y\right) \cdot x\\

\mathbf{else}:\\
\;\;\;\;y \cdot \left(y \cdot x\right)\\


\end{array}

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	5.6
Target	0.1
Herbie	0.1

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

Derivation

Split input into 2 regimes
if (*.f64 y y) < 1.99999999999999989e183
1. Initial program 0.1
  \[x \cdot \left(1 + y \cdot y\right) \]
2. Applied egg-rr0.1
  \[\leadsto \color{blue}{x \cdot \left(y \cdot y\right) + x} \]
if 1.99999999999999989e183 < (*.f64 y y)
1. Initial program 33.1
  \[x \cdot \left(1 + y \cdot y\right) \]
2. Taylor expanded in y around inf 33.2
  \[\leadsto \color{blue}{{y}^{2} \cdot x} \]
3. Simplified0.3
  \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} \]
  Proof
  (*.f64 y (*.f64 y x)): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) x)): 59 points increase in error, 31 points decrease in error
  (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 0 points increase in error, 0 points decrease in error
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+183}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2
Error	6.5
Cost	584

\[\begin{array}{l} t_0 := \left(y \cdot y\right) \cdot x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Error	20.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :herbie-target
  (+ x (* (* x y) y))

  (* x (+ 1.0 (* y y))))

Error