Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 8.13% → 0.14%

Time: 7.1s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (fma y (* y x) x))

C

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

↓

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

Julia

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

↓

function code(x, y)
	return fma(y, Float64(y * x), x)
end

Wolfram

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

↓

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

Target

Original	8.13%
Target	0.14%
Herbie	0.14%

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

Derivation

1. Initial program 8.13

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

2. Simplified 8.11

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

   [Start] 8.13	\[ x \cdot \left(1 + y \cdot y\right) \]
   distribute-lft-in [=>] 8.12	\[ \color{blue}{x \cdot 1 + x \cdot \left(y \cdot y\right)} \]
   +-commutative [=>] 8.12	\[ \color{blue}{x \cdot \left(y \cdot y\right) + x \cdot 1} \]
   *-rgt-identity [=>] 8.12	\[ x \cdot \left(y \cdot y\right) + \color{blue}{x} \]
   fma-def [=>] 8.11	\[ \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

3. Taylor expanded in x around 0 8.13

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

4. Simplified 0.14

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

   [Start] 8.13	\[ \left(1 + {y}^{2}\right) \cdot x \]
   +-commutative [=>] 8.13	\[ \color{blue}{\left({y}^{2} + 1\right)} \cdot x \]
   distribute-lft1-in [<=] 8.12	\[ \color{blue}{{y}^{2} \cdot x + x} \]
   unpow2 [=>] 8.12	\[ \color{blue}{\left(y \cdot y\right)} \cdot x + x \]
   associate-*l* [=>] 0.14	\[ \color{blue}{y \cdot \left(y \cdot x\right)} + x \]
   *-commutative [<=] 0.14	\[ y \cdot \color{blue}{\left(x \cdot y\right)} + x \]
   fma-def [=>] 0.14	\[ \color{blue}{\mathsf{fma}\left(y, x \cdot y, x\right)} \]
   *-commutative [=>] 0.14	\[ \mathsf{fma}\left(y, \color{blue}{y \cdot x}, x\right) \]

5. Final simplification 0.14

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

Alternatives

Alternative 1
Error	0.15%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2
Error	0.14%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	9.64%
Cost	585

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

Alternative 4
Error	1.67%
Cost	580

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

Alternative 5
Error	0.14%
Cost	448

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

Alternative 6
Error	32.33%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))