Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.4 → 0.1

Time: 5.5s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+73}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{1}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+153)
   (* y (* y x))
   (if (<= y 6e+73) (+ x (* x (* y y))) (/ (* 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 <= -1e+153) {
		tmp = y * (y * x);
	} else if (y <= 6e+73) {
		tmp = x + (x * (y * y));
	} else {
		tmp = (y * x) / (1.0 / y);
	}
	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+153)) then
        tmp = y * (y * x)
    else if (y <= 6d+73) then
        tmp = x + (x * (y * y))
    else
        tmp = (y * x) / (1.0d0 / y)
    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+153) {
		tmp = y * (y * x);
	} else if (y <= 6e+73) {
		tmp = x + (x * (y * y));
	} else {
		tmp = (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 <= -1e+153:
		tmp = y * (y * x)
	elif y <= 6e+73:
		tmp = x + (x * (y * y))
	else:
		tmp = (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 <= -1e+153)
		tmp = Float64(y * Float64(y * x));
	elseif (y <= 6e+73)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(Float64(y * 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 <= -1e+153)
		tmp = y * (y * x);
	elseif (y <= 6e+73)
		tmp = x + (x * (y * y));
	else
		tmp = (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, -1e+153], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+73], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(1.0 / y), $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 < -1e153

   1. Initial program 62.9

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

   2. Taylor expanded in y around inf 62.9

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} \]
      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)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 3 points increase in error, 0 points decrease in error

   if -1e153 < y < 6.00000000000000021e73

   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 6.00000000000000021e73 < y

   1. Initial program 27.9

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

   2. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + {y}^{6}\right)}{1 + \left({y}^{4} - y \cdot y\right)}} \]

   3. Simplified 64.0

      \[\leadsto \color{blue}{\frac{x}{\frac{1 + \left({y}^{4} - y \cdot y\right)}{1 + {y}^{6}}}} \]
      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)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 3 points increase in error, 0 points decrease in error

   4. Taylor expanded in y around inf 28.0

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{y}^{2}}}} \]

   5. Simplified 26.2

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{1}{y}}{y}}} \]
      Proof
      (/.f64 x (/.f64 (/.f64 1 y) y)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 y y)))): 2 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 y 2)))): 3 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.3

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

   7. Applied egg-rr 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+73}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{1}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+219}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+219}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	6.4
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 4
Error	1.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5
Error	20.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(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))))