Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.4 → 0.1

Time: 6.7s

Precision: binary64

Cost: 7176

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, y\right) \cdot \frac{x}{\frac{1}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+141)
   (* y (* x y))
   (if (<= y 2e+124) (+ x (* x (* y y))) (* (hypot 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 <= -5e+141) {
		tmp = y * (x * y);
	} else if (y <= 2e+124) {
		tmp = x + (x * (y * y));
	} else {
		tmp = hypot(1.0, y) * (x / (1.0 / y));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y) {
	double tmp;
	if (y <= -5e+141) {
		tmp = y * (x * y);
	} else if (y <= 2e+124) {
		tmp = x + (x * (y * y));
	} else {
		tmp = Math.hypot(1.0, y) * (x / (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	return x * (1.0 + (y * y))

↓

def code(x, y):
	tmp = 0
	if y <= -5e+141:
		tmp = y * (x * y)
	elif y <= 2e+124:
		tmp = x + (x * (y * y))
	else:
		tmp = math.hypot(1.0, 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 <= -5e+141)
		tmp = Float64(y * Float64(x * y));
	elseif (y <= 2e+124)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(hypot(1.0, y) * Float64(x / Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e+141)
		tmp = y * (x * y);
	elseif (y <= 2e+124)
		tmp = x + (x * (y * y));
	else
		tmp = hypot(1.0, y) * (x / (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[y, -5e+141], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+124], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision] * N[(x / N[(1.0 / y), $MachinePrecision]), $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 < -5.00000000000000025e141

   1. Initial program 54.1

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

   2. Taylor expanded in y around inf 54.1

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

   3. Simplified 0.3

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

      [Start] 54.1	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 54.1	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 0.3	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

   if -5.00000000000000025e141 < y < 1.9999999999999999e124

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 1.9999999999999999e124 < y

   1. Initial program 45.6

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

   2. Applied egg-rr 64.0

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

   3. Applied egg-rr 45.7

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

   4. Applied egg-rr 0.4

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

   5. Taylor expanded in y around inf 0.4

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{y}}} \cdot \mathsf{hypot}\left(1, y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+141}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, y\right) \cdot \frac{x}{\frac{1}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	13376

\[\mathsf{hypot}\left(1, y\right) \cdot \frac{x}{\frac{1}{\mathsf{hypot}\left(1, y\right)}} \]

Alternative 2
Error	0.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+141} \lor \neg \left(y \leq 5 \cdot 10^{+128}\right):\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \end{array} \]

Alternative 3
Error	0.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+141} \lor \neg \left(y \leq 5 \cdot 10^{+128}\right):\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 4
Error	6.2
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
Error	0.9
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
Error	20.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023054 
(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))))