Average Error: 22.7 → 14.5
Time: 5.4s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
1 - \left(1 - x\right) \cdot \frac{y}{1 + y}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y) :precision binary64 (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y)))))
double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}
↓
double code(double x, double y) {
	return 1.0 - ((1.0 - x) * (y / (1.0 + 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

Original22.7
Target0.2
Herbie14.5
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Initial program 22.7

    \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity_binary64_1203722.7

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

  4. Applied times-frac_binary64_1203214.5

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

  5. Simplified14.5

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

  6. Simplified14.5

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

  7. Final simplification14.5

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

Reproduce

herbie shell --seed 2020268 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

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

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