Average Error: 5.3 → 1.0
Time: 15.4s
Precision: binary64
\[[x, y, z, t]=\mathsf{sort}([x, y, z, t])\]
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
↓
\[\left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \]
\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}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\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)))
    (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (- (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))) + (1.0 / (sqrt(1.0 + y) + sqrt(y)))) + (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

Original5.3
Target0.3
Herbie1.0
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

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. Using strategy rm

  3. Applied flip--_binary645.2

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

  4. Simplified2.5

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

  5. Simplified2.5

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

  7. Applied flip--_binary642.3

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

  8. Simplified1.2

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

  9. Simplified1.2

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

  11. Applied flip--_binary641.2

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

  12. Simplified1.0

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

  13. Simplified1.0

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

  14. Final simplification1.0

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

Reproduce

herbie shell --seed 2021198 
(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))))