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

Average Error: 0.2 → 0.2

Time: 3.4s

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	0.2
Target	0.3
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 32.7

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

3. Simplified 32.7

   \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{y \cdot 3}}}{\sqrt{y \cdot 3}}} \]
   Proof
   (/.f64 (/.f64 x (sqrt.f64 (*.f64 y 3))) (sqrt.f64 (*.f64 y 3))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 x (sqrt.f64 (*.f64 y 3))))) (sqrt.f64 (*.f64 y 3))): 0 points increase in error, 3 points decrease in error
   (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 y 3))) (/.f64 x (sqrt.f64 (*.f64 y 3))))): 3 points increase in error, 0 points decrease in error

4. Taylor expanded in x around 0 0.8

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

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
   (/.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) 3) y): 4 points increase in error, 3 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 1 x) (*.f64 3 y))): 0 points increase in error, 4 points decrease in error
   (/.f64 (Rewrite=> *-lft-identity_binary64 x) (*.f64 3 y)): 3 points increase in error, 0 points decrease in error
   (/.f64 x (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 3) (sqrt.f64 3))) y)): 5 points increase in error, 0 points decrease in error
   (/.f64 x (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 3) 2)) y)): 2 points increase in error, 5 points decrease in error
   (/.f64 x (Rewrite<= *-commutative_binary64 (*.f64 y (pow.f64 (sqrt.f64 3) 2)))): 0 points increase in error, 2 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	0.2
Cost	320

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

Reproduce

herbie shell --seed 2022343 
(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)))