Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, C

Average Accuracy: 99.6% → 99.7%

Time: 3.6s

Precision: binary64

Cost: 320.00

Math

\[\frac{x}{y \cdot 3} \]

↓

\[\frac{\frac{x}{3}}{y} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (* y 3.0)))

↓

(FPCore (x y) :precision binary64 (/ (/ x 3.0) y))

C

double code(double x, double y) {
	return x / (y * 3.0);
}

↓

double code(double x, double y) {
	return (x / 3.0) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * 3.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / 3.0d0) / y
end function

Java

public static double code(double x, double y) {
	return x / (y * 3.0);
}

↓

public static double code(double x, double y) {
	return (x / 3.0) / y;
}

Python

def code(x, y):
	return x / (y * 3.0)

↓

def code(x, y):
	return (x / 3.0) / y

Julia

function code(x, y)
	return Float64(x / Float64(y * 3.0))
end

↓

function code(x, y)
	return Float64(Float64(x / 3.0) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * 3.0);
end

↓

function tmp = code(x, y)
	tmp = (x / 3.0) / y;
end

Wolfram

code[x_, y_] := N[(x / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(x / 3.0), $MachinePrecision] / y), $MachinePrecision]

Target

Original	99.6%
Target	99.6%
Herbie	99.7%

\[\frac{\frac{x}{y}}{3} \]

Derivation

1. Initial program 99.6%

   \[\frac{x}{y \cdot 3} \]

2. Applied egg-rr 49.5%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{y \cdot 3}} \cdot \frac{x}{\sqrt{y \cdot 3}}} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{y \cdot 3}}}{\sqrt{y \cdot 3}}} \]
   Proof

   [Start] 49.5	\[ \frac{1}{\sqrt{y \cdot 3}} \cdot \frac{x}{\sqrt{y \cdot 3}} \]
   associate-*l/ [=>] 49.5	\[ \color{blue}{\frac{1 \cdot \frac{x}{\sqrt{y \cdot 3}}}{\sqrt{y \cdot 3}}} \]
   *-lft-identity [=>] 49.5	\[ \frac{\color{blue}{\frac{x}{\sqrt{y \cdot 3}}}}{\sqrt{y \cdot 3}} \]

4. Taylor expanded in x around 0 98.7%

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\frac{x}{3}}{y}} \]
   Proof

   [Start] 98.7	\[ \frac{x}{y \cdot {\left(\sqrt{3}\right)}^{2}} \]
   *-commutative [=>] 98.7	\[ \frac{x}{\color{blue}{{\left(\sqrt{3}\right)}^{2} \cdot y}} \]
   *-lft-identity [<=] 98.7	\[ \frac{\color{blue}{1 \cdot x}}{{\left(\sqrt{3}\right)}^{2} \cdot y} \]
   unpow2 [=>] 98.7	\[ \frac{1 \cdot x}{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)} \cdot y} \]
   rem-square-sqrt [=>] 99.6	\[ \frac{1 \cdot x}{\color{blue}{3} \cdot y} \]
   associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{1 \cdot x}{3}}{y}} \]
   *-lft-identity [=>] 99.7	\[ \frac{\frac{\color{blue}{x}}{3}}{y} \]

6. Final simplification 99.7%

   \[\leadsto \frac{\frac{x}{3}}{y} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	320.00

\[x \cdot \frac{0.3333333333333333}{y} \]

Alternative 2
Accuracy	99.5%
Cost	320.00

\[\frac{x}{y} \cdot 0.3333333333333333 \]

Alternative 3
Accuracy	99.6%
Cost	320.00

\[\frac{x}{3 \cdot y} \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, C"
  :precision binary64

  :herbie-target
  (/ (/ x y) 3.0)

  (/ x (* y 3.0)))