Average Error: 5.4 → 2.8
Time: 13.4s
Precision: binary64
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
↓
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + z \cdot \sqrt{z}}} \cdot \frac{\sqrt{\left(1 + z\right) + \left(z - \sqrt{1 + z} \cdot \sqrt{z}\right)}}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\]
\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)
↓
\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + z \cdot \sqrt{z}}} \cdot \frac{\sqrt{\left(1 + z\right) + \left(z - \sqrt{1 + z} \cdot \sqrt{z}\right)}}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}
(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
   (*
    (/ 1.0 (sqrt (+ (pow (sqrt (+ 1.0 z)) 3.0) (* z (sqrt z)))))
    (/
     (sqrt (+ (+ 1.0 z) (- z (* (sqrt (+ 1.0 z)) (sqrt z)))))
     (sqrt (+ (sqrt (+ 1.0 z)) (sqrt z))))))
  (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))
double code(double x, double y, double z, double t) {
	return (((sqrt(x + 1.0) - sqrt(x)) + (sqrt(y + 1.0) - sqrt(y))) + (sqrt(z + 1.0) - sqrt(z))) + (sqrt(t + 1.0) - sqrt(t));
}
↓
double code(double x, double y, double z, double t) {
	return (((sqrt(x + 1.0) - sqrt(x)) + (sqrt(1.0 + y) - sqrt(y))) + ((1.0 / sqrt(pow(sqrt(1.0 + z), 3.0) + (z * sqrt(z)))) * (sqrt((1.0 + z) + (z - (sqrt(1.0 + z) * sqrt(z)))) / sqrt(sqrt(1.0 + z) + sqrt(z))))) + (1.0 / (sqrt(1.0 + t) + sqrt(t)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.4
Target1.8
Herbie2.8
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

Derivation

  1. Initial program 5.4

    \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied flip--_binary645.3

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

  4. Simplified4.0

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{1}}{\sqrt{t + 1} + \sqrt{t}}\]
  5. Using strategy rm

  6. Applied flip--_binary643.9

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

  7. Simplified2.7

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{1}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt_binary642.7

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

  10. Applied associate-/r*_binary642.7

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

  11. Simplified2.7

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}}}{\sqrt{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity_binary642.7

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

  14. Applied sqrt-prod_binary642.7

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

  15. Applied flip3-+_binary642.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\frac{1}{\sqrt{\color{blue}{\frac{{\left(\sqrt{1 + z}\right)}^{3} + {\left(\sqrt{z}\right)}^{3}}{\sqrt{1 + z} \cdot \sqrt{1 + z} + \left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{z}\right)}}}}}{\sqrt{1} \cdot \sqrt{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]

  16. Applied sqrt-div_binary642.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\frac{1}{\color{blue}{\frac{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + {\left(\sqrt{z}\right)}^{3}}}{\sqrt{\sqrt{1 + z} \cdot \sqrt{1 + z} + \left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{z}\right)}}}}}{\sqrt{1} \cdot \sqrt{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]

  17. Applied associate-/r/_binary642.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + {\left(\sqrt{z}\right)}^{3}}} \cdot \sqrt{\sqrt{1 + z} \cdot \sqrt{1 + z} + \left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{z}\right)}}}{\sqrt{1} \cdot \sqrt{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]

  18. Applied times-frac_binary642.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + {\left(\sqrt{z}\right)}^{3}}}}{\sqrt{1}} \cdot \frac{\sqrt{\sqrt{1 + z} \cdot \sqrt{1 + z} + \left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{z}\right)}}{\sqrt{\sqrt{z + 1} + \sqrt{z}}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]

  19. Simplified2.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + z \cdot \sqrt{z}}}} \cdot \frac{\sqrt{\sqrt{1 + z} \cdot \sqrt{1 + z} + \left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{z}\right)}}{\sqrt{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]

  20. Simplified2.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + z \cdot \sqrt{z}}} \cdot \color{blue}{\frac{\sqrt{\left(1 + z\right) + \left(z - \sqrt{1 + z} \cdot \sqrt{z}\right)}}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}}\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]

  21. Final simplification2.8

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{{\left(\sqrt{1 + z}\right)}^{3} + z \cdot \sqrt{z}}} \cdot \frac{\sqrt{\left(1 + z\right) + \left(z - \sqrt{1 + z} \cdot \sqrt{z}\right)}}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :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))))