Main:z from

Average Error: 5.2 → 0.1

Time: 20.4s

Precision: binary64

Cost: 53184

Math

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

↓

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

FPCore

(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)))))

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 + 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)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.2
Target	1.3
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	5.2
Cost	98240

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

Alternative 2
Error	5.2
Cost	98240

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

Alternative 3
Error	5.2
Cost	98240

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

Alternative 4
Error	2.6
Cost	79040

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

Alternative 5
Error	5.2
Cost	78656

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

Alternative 6
Error	5.2
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 7
Error	22.5
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 8
Error	22.1
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 9
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 10
Error	22.5
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 11
Error	1.4
Cost	65920

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

Alternative 12
Error	3.9
Cost	65664

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

Alternative 13
Error	5.2
Cost	65536

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

Alternative 14
Error	5.2
Cost	65536

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

Alternative 15
Error	5.2
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 16
Error	5.2
Cost	65472

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

Alternative 17
Error	5.2
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 18
Error	0.1
Cost	53184

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

Alternative 19
Error	1.3
Cost	53056

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

Alternative 20
Error	2.7
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 21
Error	2.7
Cost	52928

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

Alternative 22
Error	2.6
Cost	52928

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

Alternative 23
Error	3.9
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 24
Error	3.8
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 25
Error	4.0
Cost	52800

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

Alternative 26
Error	4.0
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 27
Error	5.2
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 28
Error	27.6
Cost	40000

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

Alternative 29
Error	27.8
Cost	39872

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

Alternative 30
Error	27.9
Cost	39872

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

Alternative 31
Error	27.1
Cost	39872

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

Alternative 32
Error	28.1
Cost	39872

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

Alternative 33
Error	28.3
Cost	39744

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

Alternative 34
Error	28.0
Cost	39744

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

Alternative 35
Error	28.1
Cost	39744

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

Alternative 36
Error	28.2
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 37
Error	27.6
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 38
Error	28.5
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 39
Error	28.3
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 40
Error	42.0
Cost	64

\[1\]

Alternative 41
Error	62.0
Cost	64

\[0\]

Alternative 42
Error	63.0
Cost	64

\[-1\]

Derivation

1. Initial program 5.2

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

3. Applied flip--_binary64_13699 5.1

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

4. Simplified 3.8

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

5. Simplified 3.8

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

7. Applied flip--_binary64_13699 3.7

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

8. Simplified 2.6

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

9. Simplified 2.6

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

11. Applied flip--_binary64_13699 2.5

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

12. Simplified 1.3

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

14. Applied flip--_binary64_13699 1.3

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

15. Simplified 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2021042 
(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))))