Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.4 → 0.1

Time: 5.6s

Precision: binary64

Cost: 6984

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{1}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+52)
   (* y (* y x))
   (if (<= y 3.2e+133) (* x (fma y y 1.0)) (/ (* y x) (/ 1.0 y)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (y <= -8e+52) {
		tmp = y * (y * x);
	} else if (y <= 3.2e+133) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = (y * x) / (1.0 / y);
	}
	return tmp;
}

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (y <= -8e+52)
		tmp = Float64(y * Float64(y * x));
	elseif (y <= 3.2e+133)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(Float64(y * x) / Float64(1.0 / y));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[y, -8e+52], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+133], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	5.4
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -7.9999999999999999e52

   1. Initial program 23.2

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

   2. Taylor expanded in y around inf 23.2

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

   3. Simplified 0.3

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

      [Start] 23.2	\[ {y}^{2} \cdot x \]
      *-commutative [=>] 23.2	\[ \color{blue}{x \cdot {y}^{2}} \]
      unpow2 [=>] 23.2	\[ x \cdot \color{blue}{\left(y \cdot y\right)} \]
      associate-*r* [=>] 0.3	\[ \color{blue}{\left(x \cdot y\right) \cdot y} \]

   if -7.9999999999999999e52 < y < 3.19999999999999997e133

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ x \cdot \left(1 + y \cdot y\right) \]
      +-commutative [=>] 0.1	\[ x \cdot \color{blue}{\left(y \cdot y + 1\right)} \]
      fma-def [=>] 0.1	\[ x \cdot \color{blue}{\mathsf{fma}\left(y, y, 1\right)} \]

   if 3.19999999999999997e133 < y

   1. Initial program 51.4

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

   2. Applied egg-rr 64.0

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

   3. Simplified 64.0

      \[\leadsto \color{blue}{\frac{x}{\frac{1 - y \cdot y}{1 - {y}^{4}}}} \]
      Proof

      [Start] 64.0	\[ \frac{x \cdot \left(1 - {y}^{4}\right)}{1 - y \cdot y} \]
      associate-/l* [=>] 64.0	\[ \color{blue}{\frac{x}{\frac{1 - y \cdot y}{1 - {y}^{4}}}} \]

   4. Taylor expanded in y around inf 51.4

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

   5. Simplified 48.7

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{1}{y}}{y}}} \]
      Proof

      [Start] 51.4	\[ \frac{x}{\frac{1}{{y}^{2}}} \]
      unpow2 [=>] 51.4	\[ \frac{x}{\frac{1}{\color{blue}{y \cdot y}}} \]
      associate-/r* [=>] 48.7	\[ \frac{x}{\color{blue}{\frac{\frac{1}{y}}{y}}} \]

   6. Applied egg-rr 0.4

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

   7. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{1}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{1}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -98000000 \lor \neg \left(y \leq 1.6 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -95000000 \lor \neg \left(y \leq 10^{+19}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 3
Error	0.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -126000000:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 10^{+132}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{1}{y}}\\ \end{array} \]

Alternative 4
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 5
Error	20.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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))))