Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.7% → 99.9%

Time: 9.5s

Precision: binary64

Cost: 6980

• 26.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+207) (fma x (* y y) x) (* y (* x y))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+207) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = y * (x * 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 (Float64(y * y) <= 5e+207)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = Float64(y * Float64(x * 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[N[(y * y), $MachinePrecision], 5e+207], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Target

Original	94.7%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999999e207

   1. Initial program 99.9%

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

   2. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ x \cdot \left(1 + y \cdot y\right) \]
      +-commutative [=>] 99.9%	\[ x \cdot \color{blue}{\left(y \cdot y + 1\right)} \]
      distribute-lft-in [=>] 99.9%	\[ \color{blue}{x \cdot \left(y \cdot y\right) + x \cdot 1} \]
      *-rgt-identity [=>] 99.9%	\[ x \cdot \left(y \cdot y\right) + \color{blue}{x} \]
      fma-def [=>] 99.9%	\[ \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

   if 4.9999999999999999e207 < (*.f64 y y)

   1. Initial program 85.3%

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

   2. Taylor expanded in y around inf 85.3%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} \]
      Step-by-step derivation

      [Start] 85.3%	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 85.3%	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 99.9%	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	26112

\[\mathsf{fma}\left(\left(x \cdot y\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y}, x\right) \]

Alternative 3
Accuracy	99.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+207}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4
Accuracy	93.4%
Cost	580

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

Alternative 5
Accuracy	98.6%
Cost	580

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

Alternative 6
Accuracy	51.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(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))))