Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Accuracy: 91.4% → 99.8%

Time: 3.4s

Precision: binary64

Cost: 20104

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(y \cdot x\right)}^{2}} \cdot \left(y \cdot \sqrt[3]{y \cdot x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2e+40)
   (* y (* y x))
   (if (<= y 1.06e+152)
     (* x (+ 1.0 (* y y)))
     (* (cbrt (pow (* y x) 2.0)) (* y (cbrt (* 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 <= -6.2e+40) {
		tmp = y * (y * x);
	} else if (y <= 1.06e+152) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = cbrt(pow((y * x), 2.0)) * (y * cbrt((y * x)));
	}
	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 <= -6.2e+40) {
		tmp = y * (y * x);
	} else if (y <= 1.06e+152) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = Math.cbrt(Math.pow((y * x), 2.0)) * (y * Math.cbrt((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 (y <= -6.2e+40)
		tmp = Float64(y * Float64(y * x));
	elseif (y <= 1.06e+152)
		tmp = Float64(x * Float64(1.0 + Float64(y * y)));
	else
		tmp = Float64(cbrt((Float64(y * x) ^ 2.0)) * Float64(y * cbrt(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[y, -6.2e+40], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e+152], N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(y * x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(y * N[Power[N[(y * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	91.4%
Target	99.9%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if y < -6.1999999999999995e40

   1. Initial program 65.6%

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

   2. Taylor expanded in y around inf 65.6%

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

   3. Simplified 99.6%

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

      [Start] 65.6	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 65.6	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 99.6	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

   if -6.1999999999999995e40 < y < 1.06e152

   1. Initial program 99.9%

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

   if 1.06e152 < y

   1. Initial program 2.4%

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

   2. Taylor expanded in y around inf 2.4%

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

   3. Simplified 99.5%

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

      [Start] 2.4	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 2.4	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 99.5	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

   4. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(y \cdot x\right)}\right)}^{3}} \]
      Proof

      [Start] 99.5	\[ y \cdot \left(y \cdot x\right) \]
      add-cube-cbrt [=>] 98.1	\[ \color{blue}{\left(\sqrt[3]{y \cdot \left(y \cdot x\right)} \cdot \sqrt[3]{y \cdot \left(y \cdot x\right)}\right) \cdot \sqrt[3]{y \cdot \left(y \cdot x\right)}} \]
      pow3 [=>] 98.1	\[ \color{blue}{{\left(\sqrt[3]{y \cdot \left(y \cdot x\right)}\right)}^{3}} \]

   5. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{2} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right)} \]
      Proof

      [Start] 98.1	\[ {\left(\sqrt[3]{y \cdot \left(y \cdot x\right)}\right)}^{3} \]
      rem-cube-cbrt [=>] 99.5	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]
      *-commutative [=>] 99.5	\[ \color{blue}{\left(y \cdot x\right) \cdot y} \]
      add-cube-cbrt [=>] 97.9	\[ \color{blue}{\left(\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right) \cdot \sqrt[3]{y \cdot x}\right)} \cdot y \]
      associate-*l* [=>] 97.9	\[ \color{blue}{\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right) \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right)} \]
      pow2 [=>] 97.9	\[ \color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{2}} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right) \]

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(y \cdot x\right)}^{2}}} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right) \]
      Proof

      [Start] 97.9	\[ {\left(\sqrt[3]{y \cdot x}\right)}^{2} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right) \]
      unpow2 [=>] 97.9	\[ \color{blue}{\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right)} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right) \]
      cbrt-unprod [=>] 98.5	\[ \color{blue}{\sqrt[3]{\left(y \cdot x\right) \cdot \left(y \cdot x\right)}} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right) \]
      pow2 [=>] 98.5	\[ \sqrt[3]{\color{blue}{{\left(y \cdot x\right)}^{2}}} \cdot \left(\sqrt[3]{y \cdot x} \cdot y\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(y \cdot x\right)}^{2}} \cdot \left(y \cdot \sqrt[3]{y \cdot x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+41}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y} \cdot \left(\left(y \cdot x\right) \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5000000000000:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Accuracy	90.0%
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 4
Accuracy	97.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5
Accuracy	67.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(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))))