Average Error: 5.7 → 1.6
Time: 7.8m
Precision: 64
Internal Precision: 1408
\[\left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \left(\sqrt{y + 1.0} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1.0} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1.0} - \sqrt{t}\right)\]
↓
\[\left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \frac{1.0}{\sqrt{y + 1.0} + \sqrt{y}}\right) + \frac{1.0}{\sqrt{z + 1.0} + \sqrt{z}}\right) + \frac{1.0}{\sqrt{t + 1.0} + \sqrt{t}}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.7
Target1.7
Herbie1.6
\[\left(\left(\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}} + \frac{1.0}{\sqrt{y + 1.0} + \sqrt{y}}\right) + \frac{1.0}{\sqrt{z + 1.0} + \sqrt{z}}\right) + \left(\sqrt{t + 1.0} - \sqrt{t}\right)\]

Derivation

  1. Initial program 5.7

    \[\left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \left(\sqrt{y + 1.0} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1.0} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1.0} - \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied flip--5.6

    \[\leadsto \left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \left(\sqrt{y + 1.0} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1.0} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1.0} \cdot \sqrt{t + 1.0} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1.0} + \sqrt{t}}}\]

  4. Applied simplify4.3

    \[\leadsto \left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \left(\sqrt{y + 1.0} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1.0} - \sqrt{z}\right)\right) + \frac{\color{blue}{1.0}}{\sqrt{t + 1.0} + \sqrt{t}}\]
  5. Using strategy rm

  6. Applied flip--4.2

    \[\leadsto \left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \left(\sqrt{y + 1.0} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1.0} \cdot \sqrt{z + 1.0} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1.0} + \sqrt{z}}}\right) + \frac{1.0}{\sqrt{t + 1.0} + \sqrt{t}}\]

  7. Applied simplify3.0

    \[\leadsto \left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \left(\sqrt{y + 1.0} - \sqrt{y}\right)\right) + \frac{\color{blue}{1.0}}{\sqrt{z + 1.0} + \sqrt{z}}\right) + \frac{1.0}{\sqrt{t + 1.0} + \sqrt{t}}\]
  8. Using strategy rm

  9. Applied flip--2.9

    \[\leadsto \left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1.0} \cdot \sqrt{y + 1.0} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1.0} + \sqrt{y}}}\right) + \frac{1.0}{\sqrt{z + 1.0} + \sqrt{z}}\right) + \frac{1.0}{\sqrt{t + 1.0} + \sqrt{t}}\]

  10. Applied simplify1.6

    \[\leadsto \left(\left(\left(\sqrt{x + 1.0} - \sqrt{x}\right) + \frac{\color{blue}{1.0}}{\sqrt{y + 1.0} + \sqrt{y}}\right) + \frac{1.0}{\sqrt{z + 1.0} + \sqrt{z}}\right) + \frac{1.0}{\sqrt{t + 1.0} + \sqrt{t}}\]
  11. Removed slow pow expressions.

Runtime

Time bar (total: 7.8m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(FPCore (x y z t)
  :name "Main:z from "

  :herbie-target
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))