Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.4 → 0.1

Time: 6.9s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 100000000000.0) (fma x (* y y) x) (* y (* y x))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 100000000000.0) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = y * (y * x);
	}
	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 (Float64(y * y) <= 100000000000.0)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 100000000000.0], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(y * x), $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 2 regimes

2. if (*.f64 y y) < 1e11

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ x \cdot \left(1 + y \cdot y\right) \]
      distribute-lft-in [=>] 0.0	\[ \color{blue}{x \cdot 1 + x \cdot \left(y \cdot y\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{x \cdot \left(y \cdot y\right) + x \cdot 1} \]
      *-rgt-identity [=>] 0.0	\[ x \cdot \left(y \cdot y\right) + \color{blue}{x} \]
      fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

   if 1e11 < (*.f64 y y)

   1. Initial program 16.2

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

   2. Taylor expanded in y around inf 16.4

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

   3. Simplified 0.4

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

      [Start] 16.4	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 16.4	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 0.4	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 100000000000:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2
Error	6.2
Cost	580

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

Alternative 3
Error	0.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Error	21.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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))))