Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.4 → 0.1

Time: 1.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y x))))
   (if (<= y -1e+59) t_0 (if (<= y 1e+128) (fma x (* y y) x) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = y * (y * x);
	double tmp;
	if (y <= -1e+59) {
		tmp = t_0;
	} else if (y <= 1e+128) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x, y)
	t_0 = Float64(y * Float64(y * x))
	tmp = 0.0
	if (y <= -1e+59)
		tmp = t_0;
	elseif (y <= 1e+128)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e+59], t$95$0, If[LessEqual[y, 1e+128], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Target

Original	5.4
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999972e58 or 1.0000000000000001e128 < y

   1. Initial program 32.7

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

   2. Simplified 32.7

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

   3. Taylor expanded in y around inf 32.7

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

   4. Simplified 0.3

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

   if -9.99999999999999972e58 < y < 1.0000000000000001e128

   1. Initial program 0.1

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

   2. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+59}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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))))