Average Error: 5.4 → 2.9
Time: 11.2s
Precision: binary64
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]
↓
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{1}{\sqrt{t + 1} + \sqrt{t}}\]
\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{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{1}{\sqrt{t + 1} + \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) (((double) (1.0 / ((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) (1.0 / ((double) (((double) sqrt(((double) (t + 1.0)))) + ((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
Herbie2.9
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\]

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--5.4

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

  4. Simplified5.1

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

  5. Taylor expanded around 0 4.2

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

  7. Applied flip--4.1

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

  8. Simplified3.9

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

  9. Taylor expanded around 0 2.9

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

  10. Final simplification2.9

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

Reproduce

herbie shell --seed 2020169 
(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))))