Average Error: 5.4 → 0.1
Time: 9.9s
Precision: binary64
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
↓
\[\left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\]
\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}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\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)))
    (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (/ 1.0 (+ (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 ((double) (((double) (((double) (((double) (((double) sqrt(((double) (x + 1.0)))) - ((double) sqrt(x)))) + ((double) (((double) sqrt(((double) (y + 1.0)))) - ((double) sqrt(y)))))) + ((double) (((double) sqrt(((double) (z + 1.0)))) - ((double) sqrt(z)))))) + ((double) (((double) sqrt(((double) (t + 1.0)))) - ((double) sqrt(t))))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) ((1.0 / ((double) (((double) sqrt(((double) (1.0 + x)))) + ((double) sqrt(x))))) + (1.0 / ((double) (((double) sqrt(((double) (1.0 + y)))) + ((double) sqrt(y))))))) + (1.0 / ((double) (((double) sqrt(((double) (1.0 + z)))) + ((double) sqrt(z))))))) + (1.0 / ((double) (((double) sqrt(((double) (1.0 + t)))) + ((double) 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
Target1.5
Herbie0.1
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

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--5.3

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

  4. Simplified4.2

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{1 + 0}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

  5. Simplified4.2

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1 + 0}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
  6. Using strategy rm

  7. Applied flip--4.1

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1 + 0}{\sqrt{1 + y} + \sqrt{y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

  8. Simplified2.9

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1 + 0}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{\color{blue}{1}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
  9. Using strategy rm

  10. Applied flip--2.8

    \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{1 + 0}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

  11. Simplified1.5

    \[\leadsto \left(\left(\frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} + \frac{1 + 0}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
  12. Using strategy rm

  13. Applied flip--1.4

    \[\leadsto \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1 + 0}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}}\]

  14. Simplified0.1

    \[\leadsto \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1 + 0}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\color{blue}{1}}{\sqrt{t + 1} + \sqrt{t}}\]

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020198 
(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))))