Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.3 → 0.1

Time: 8.8s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+136}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+38}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{y}}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+136)
   (* y (* x y))
   (if (<= y 5e+38) (+ x (* x (* y y))) (/ y (/ (/ 1.0 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 <= -1e+136) {
		tmp = y * (x * y);
	} else if (y <= 5e+38) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y / ((1.0 / y) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1d+136)) then
        tmp = y * (x * y)
    else if (y <= 5d+38) then
        tmp = x + (x * (y * y))
    else
        tmp = y / ((1.0d0 / y) / x)
    end if
    code = tmp
end function

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 <= -1e+136) {
		tmp = y * (x * y);
	} else if (y <= 5e+38) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y / ((1.0 / y) / x);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if y <= -1e+136:
		tmp = y * (x * y)
	elif y <= 5e+38:
		tmp = x + (x * (y * y))
	else:
		tmp = y / ((1.0 / 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 <= -1e+136)
		tmp = Float64(y * Float64(x * y));
	elseif (y <= 5e+38)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(y / Float64(Float64(1.0 / y) / x));
	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 <= -1e+136)
		tmp = y * (x * y);
	elseif (y <= 5e+38)
		tmp = x + (x * (y * y));
	else
		tmp = y / ((1.0 / y) / x);
	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, -1e+136], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+38], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	5.3
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000006e136

   1. Initial program 51.3

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

   2. Taylor expanded in y around inf 51.4

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

   3. Simplified 51.3

      \[\leadsto \color{blue}{x \cdot \left(y \cdot y\right)} \]
      Proof
      (*.f64 x (*.f64 y y)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 x y) y)): 28 points increase in error, 64 points decrease in error
      (*.f64 (*.f64 x (Rewrite<= remove-double-div_binary64 (/.f64 1 (/.f64 1 y)))) y): 13 points increase in error, 5 points decrease in error
      (*.f64 (*.f64 x (Rewrite<= unpow-1_binary64 (pow.f64 (/.f64 1 y) -1))) y): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 x (pow.f64 (/.f64 1 y) (Rewrite<= metadata-eval (/.f64 -2 2)))) y): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 x (pow.f64 (/.f64 1 y) (/.f64 -2 2))) (Rewrite<= remove-double-div_binary64 (/.f64 1 (/.f64 1 y)))): 15 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 x (pow.f64 (/.f64 1 y) (/.f64 -2 2))) (Rewrite<= unpow-1_binary64 (pow.f64 (/.f64 1 y) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 x (pow.f64 (/.f64 1 y) (/.f64 -2 2))) (pow.f64 (/.f64 1 y) (Rewrite<= metadata-eval (/.f64 -2 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 x (*.f64 (pow.f64 (/.f64 1 y) (/.f64 -2 2)) (pow.f64 (/.f64 1 y) (/.f64 -2 2))))): 64 points increase in error, 28 points decrease in error
      (*.f64 x (Rewrite<= sqr-pow_binary64 (pow.f64 (/.f64 1 y) -2))): 32 points increase in error, 21 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (/.f64 1 y) -2) x)): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around 0 51.3

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

   5. Simplified 0.2

      \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} \]
      Proof
      (*.f64 y (*.f64 y x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) x)): 64 points increase in error, 28 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 0 points increase in error, 0 points decrease in error

   if -1.00000000000000006e136 < y < 4.9999999999999997e38

   1. Initial program 0.1

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

   2. Applied egg-rr 0.0

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

   if 4.9999999999999997e38 < y

   1. Initial program 21.4

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

   2. Applied egg-rr 38.0

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

   3. Simplified 21.5

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(y, y, 1\right)}}} \]
      Proof
      (/.f64 x (/.f64 1 (fma.f64 y y 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (fma.f64 y y 1)) 1)): 4 points increase in error, 12 points decrease in error
      (/.f64 (*.f64 x (fma.f64 y y 1)) (Rewrite<= *-inverses_binary64 (/.f64 (-.f64 x (*.f64 x (*.f64 y y))) (-.f64 x (*.f64 x (*.f64 y y)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 x (fma.f64 y y 1)) (-.f64 x (*.f64 x (*.f64 y y)))) (-.f64 x (*.f64 x (*.f64 y y))))): 124 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (fma.f64 y y 1) x)) (-.f64 x (*.f64 x (*.f64 y y)))) (-.f64 x (*.f64 x (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 y y) 1)) x) (-.f64 x (*.f64 x (*.f64 y y)))) (-.f64 x (*.f64 x (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 (*.f64 y y) x))) (-.f64 x (*.f64 x (*.f64 y y)))) (-.f64 x (*.f64 x (*.f64 y y)))): 2 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 x (Rewrite<= *-commutative_binary64 (*.f64 x (*.f64 y y)))) (-.f64 x (*.f64 x (*.f64 y y)))) (-.f64 x (*.f64 x (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 x x) (*.f64 (*.f64 x (*.f64 y y)) (*.f64 x (*.f64 y y))))) (-.f64 x (*.f64 x (*.f64 y y)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y around inf 21.0

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

   5. Applied egg-rr 0.4

      \[\leadsto \color{blue}{y \cdot \frac{x}{\frac{1}{y}}} \]

   6. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{y}{\frac{\frac{1}{y}}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+136}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+38}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{y}}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+123}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+123}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 3
Error	6.0
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
Error	0.8
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 5
Error	0.1
Cost	576

\[x + \frac{y}{\frac{\frac{1}{y}}{x}} \]

Alternative 6
Error	0.1
Cost	576

\[x + \frac{y}{\frac{\frac{1}{x}}{y}} \]

Alternative 7
Error	20.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))))