Average Error: 5.3 → 2.8
Time: 15.9s
Precision: binary64
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
↓
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
\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) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)

(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)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
  (- (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)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + (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

Original5.3
Target1.5
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 5.3

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

  2. Applied egg-rr5.0

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

  3. Applied egg-rr4.7

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

  4. Taylor expanded in y around 0 3.8

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

  5. Taylor expanded in z around 0 2.8

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

  6. Final simplification2.8

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

Reproduce

herbie shell --seed 2022130 
(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))))