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

Average Error: 0.3 → 0.2

Time: 2.0s

Precision: binary64

Cost: 320

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	0.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.3

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

3. Applied egg-rr 0.2

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

4. Taylor expanded in x around 0 0.4

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{x}{y}} \]

5. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\frac{x}{3}}{y}} \]
   Proof
   (/.f64 (/.f64 x 3) y): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 y 3))): 39 points increase in error, 37 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1)) (*.f64 y 3)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 x y) (/.f64 1 3))): 59 points increase in error, 33 points decrease in error
   (*.f64 (/.f64 x y) (Rewrite=> metadata-eval 1/3)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 1/3 (/.f64 x y))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	320

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

Alternative 2
Error	0.3
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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)))