Main:z from

Average Error: 5.6 → 0.1

Time: 16.4s

Precision: binary64

Cost: 53184

Math

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

↓

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

FPCore

(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 t)) (sqrt t)))
  (+
   (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
    (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))))

C

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 + t) + sqrt(t))) + ((1.0 / (sqrt(1.0 + z) + sqrt(z))) + ((1.0 / (sqrt(1.0 + y) + sqrt(y))) + (1.0 / (sqrt(1.0 + x) + sqrt(x)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.6
Target	1.5
Herbie	0.1

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

Alternatives

Alternative 1
Error	3.1
Cost	99072

\[\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{1}{{\left(\sqrt{1 + z}\right)}^{3} + z \cdot \sqrt{z}} \cdot \left(\sqrt{1 + z} \cdot \sqrt{1 + z} + \left(\sqrt{z} \cdot \sqrt{z} - \sqrt{1 + z} \cdot \sqrt{z}\right)\right)\right)\]

Alternative 2
Error	5.7
Cost	91456

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

Alternative 3
Error	5.6
Cost	78656

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

Alternative 4
Error	22.2
Cost	72832

\[\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \frac{{\left(\sqrt{1 + z}\right)}^{3} - z \cdot \sqrt{z}}{\left(1 + z\right) + \left(z + \sqrt{1 + z} \cdot \sqrt{z}\right)}\right)\]

Alternative 5
Error	22.6
Cost	72832

\[\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{{\left(\sqrt{x + 1}\right)}^{3} - x \cdot \sqrt{x}}{\left(x + 1\right) + \left(x + \sqrt{x + 1} \cdot \sqrt{x}\right)}\right)\right)\]

Alternative 6
Error	22.6
Cost	72832

\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \frac{{\left(\sqrt{1 + t}\right)}^{3} - t \cdot \sqrt{t}}{\left(1 + t\right) + \left(t + \sqrt{1 + t} \cdot \sqrt{t}\right)}\]

Alternative 7
Error	22.4
Cost	72832

\[\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{{\left(\sqrt{1 + y}\right)}^{3} - y \cdot \sqrt{y}}{\left(1 + y\right) + \left(y + \sqrt{1 + y} \cdot \sqrt{y}\right)}\right)\right)\]

Alternative 8
Error	1.7
Cost	65920

\[\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt[3]{{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)}^{3}}\right)\right)\]

Alternative 9
Error	5.6
Cost	65536

\[\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \sqrt[3]{{\left(\sqrt{x + 1} - \sqrt{x}\right)}^{3}}\right)\right)\]

Alternative 10
Error	5.6
Cost	65472

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

Alternative 11
Error	5.6
Cost	65472

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

Alternative 12
Error	5.6
Cost	65472

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

Alternative 13
Error	5.6
Cost	65472

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

Alternative 14
Error	0.1
Cost	53184

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

Alternative 15
Error	1.7
Cost	53056

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

Alternative 16
Error	2.9
Cost	52928

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

Alternative 17
Error	3.1
Cost	52928

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

Alternative 18
Error	4.3
Cost	52800

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

Alternative 19
Error	4.4
Cost	52800

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

Alternative 20
Error	4.3
Cost	52800

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

Alternative 21
Error	4.3
Cost	52800

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

Alternative 22
Error	5.6
Cost	52672

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

Alternative 23
Error	28.0
Cost	40000

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

Alternative 24
Error	28.2
Cost	39872

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

Alternative 25
Error	28.3
Cost	39872

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

Alternative 26
Error	27.7
Cost	39872

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

Alternative 27
Error	27.8
Cost	39872

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

Alternative 28
Error	28.5
Cost	39744

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

Alternative 29
Error	28.1
Cost	39744

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

Alternative 30
Error	28.0
Cost	39744

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

Alternative 31
Error	28.5
Cost	39744

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

Alternative 32
Error	28.3
Cost	39616

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

Alternative 33
Error	28.2
Cost	39616

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

Alternative 34
Error	28.7
Cost	39616

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

Alternative 35
Error	28.7
Cost	39616

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

Alternative 36
Error	41.9
Cost	64

\[1\]

Alternative 37
Error	62.0
Cost	64

\[0\]

Alternative 38
Error	63.0
Cost	64

\[-1\]

Derivation

1. Initial program 5.6

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

3. Applied flip--_binary64_11306 5.5

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

4. Simplified 4.3

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

5. Simplified 4.3

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

7. Applied flip--_binary64_11306 4.2

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

8. Simplified 3.1

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

10. Applied flip--_binary64_11306 3.0

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

11. Simplified 1.7

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

13. Applied flip--_binary64_11306 1.6

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

14. Simplified 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2021022 
(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))))