Average Error: 8.2 → 0.1
Time: 1.7s
Precision: binary64
\[\frac{x \cdot y}{y + 1} \]
↓
\[\frac{x}{1 + \frac{1}{y}} \]
(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))
↓
(FPCore (x y) :precision binary64 (/ x (+ 1.0 (/ 1.0 y))))
double code(double x, double y) {
	return (x * y) / (y + 1.0);
}
↓
double code(double x, double y) {
	return x / (1.0 + (1.0 / y));
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (y + 1.0d0)
end function
↓
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (1.0d0 + (1.0d0 / y))
end function

public static double code(double x, double y) {
	return (x * y) / (y + 1.0);
}
↓
public static double code(double x, double y) {
	return x / (1.0 + (1.0 / y));
}

def code(x, y):
	return (x * y) / (y + 1.0)
↓
def code(x, y):
	return x / (1.0 + (1.0 / y))

function code(x, y)
	return Float64(Float64(x * y) / Float64(y + 1.0))
end
↓
function code(x, y)
	return Float64(x / Float64(1.0 + Float64(1.0 / y)))
end

function tmp = code(x, y)
	tmp = (x * y) / (y + 1.0);
end
↓
function tmp = code(x, y)
	tmp = x / (1.0 + (1.0 / y));
end

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(x / N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x \cdot y}{y + 1}
↓
\frac{x}{1 + \frac{1}{y}}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.2
Target0.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

  1. Initial program 8.2

    \[\frac{x \cdot y}{y + 1} \]

  2. Applied associate-/l*_binary640.1

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

  3. Taylor expanded in y around 0 0.1

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

  4. Final simplification0.1

    \[\leadsto \frac{x}{1 + \frac{1}{y}} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))