Average Error: 5.2 → 1.6
Time: 7.9s
Precision: binary64
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
↓
\[\left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{\frac{1}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\]
\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(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{\frac{1}{\sqrt{\sqrt{1 + z} + \sqrt{z}}}}{\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
 (+
  (+
   (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (- (sqrt (+ 1.0 y)) (sqrt y)))
   (/
    (/ 1.0 (sqrt (+ (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 (((1.0 / (sqrt(1.0 + x) + sqrt(x))) + (sqrt(1.0 + y) - sqrt(y))) + ((1.0 / sqrt(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

Original5.2
Target1.4
Herbie1.6
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

Derivation

  1. Initial program 5.2

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

  3. Applied flip--_binary64_61185.1

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

  4. Simplified4.1

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

  6. Applied flip--_binary64_61184.0

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

  7. Simplified3.0

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

  9. Applied flip--_binary64_61182.9

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

  10. Simplified1.6

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

  12. Applied add-sqr-sqrt_binary64_61641.6

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

  13. Applied associate-/r*_binary64_60891.6

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

  14. Simplified1.6

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

  15. Final simplification1.6

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

Reproduce

herbie shell --seed 2020281 
(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))))