Average Error: 10.3 → 0.2
Time: 4.0s
Precision: binary64
Cost: 320
\[\frac{x}{y \cdot y} \]
↓
\[\frac{\frac{x}{y}}{y} \]
(FPCore (x y) :precision binary64 (/ x (* y y)))
↓
(FPCore (x y) :precision binary64 (/ (/ x y) y))
double code(double x, double y) {
	return x / (y * y);
}
↓
double code(double x, double y) {
	return (x / y) / y;
}

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

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

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

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

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

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

\frac{x}{y \cdot y}
↓
\frac{\frac{x}{y}}{y}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.3
Target0.2
Herbie0.2
\[\frac{\frac{x}{y}}{y} \]

Derivation

  1. Initial program 10.3

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

  2. Applied egg-rr10.5

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

  3. Taylor expanded in y around 0 10.3

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

  4. Simplified0.2

    \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
    Proof
    (/.f64 (/.f64 x y) y): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 y y))): 65 points increase in error, 23 points decrease in error
    (/.f64 x (Rewrite<= unpow2_binary64 (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Physics.ForceLayout:coulombForce from force-layout-0.4.0.2"
  :precision binary64

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

  (/ x (* y y)))