Main:z from

Average Accuracy: 91.6% → 99.9%

Time: 38.1s

Precision: binary64

Cost: 59520

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

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 x) (hypot 1.0 (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(x) + hypot(1.0, 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))));
}

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

↓

public static double code(double x, double y, double z, double t) {
	return (1.0 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x)))) + (((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z)))) + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

↓

def code(x, y, z, t):
	return (1.0 / (math.sqrt(x) + math.hypot(1.0, math.sqrt(x)))) + (((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

↓

function code(x, y, z, t)
	return Float64(Float64(1.0 / Float64(sqrt(x) + hypot(1.0, sqrt(x)))) + Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

↓

function tmp = code(x, y, z, t)
	tmp = (1.0 / (sqrt(x) + hypot(1.0, 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))));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	91.6%
Target	97.6%
Herbie	99.9%

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

Derivation

1. Initial program 90.1%

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

2. Simplified 90.1%

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

   [Start] 90.1	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 90.1	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 90.1	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
   sub-neg [=>] 90.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 90.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [<=] 90.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 90.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 90.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

3. Applied egg-rr 92.3%

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

   [Start] 90.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 90.1	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   div-inv [=>] 90.1	\[ \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 68.7	\[ \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 68.7	\[ \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 90.3	\[ \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--l+ [=>] 92.3	\[ \color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 92.3	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [=>] 92.3	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   hypot-1-def [=>] 92.3	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)} + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

4. Simplified 92.3%

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

   [Start] 92.3	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 92.3	\[ \left(1 + \color{blue}{0}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 92.3	\[ \color{blue}{1} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-lft-identity [=>] 92.3	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 92.3	\[ \frac{1}{\color{blue}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

5. Applied egg-rr 92.7%

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

   [Start] 92.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 92.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 73.9	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 92.7	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Simplified 94.5%

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

   [Start] 92.7	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--l+ [=>] 94.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 94.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 94.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{\color{blue}{1}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

7. Applied egg-rr 97.5%

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

   [Start] 94.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 94.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   div-inv [=>] 94.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 77.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 95.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--l+ [=>] 97.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

8. Simplified 97.5%

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

   [Start] 97.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 97.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 97.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-lft-identity [=>] 97.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

9. Applied egg-rr 99.8%

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

   [Start] 97.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 97.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
   div-inv [=>] 97.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
   add-sqr-sqrt [<=] 74.0	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
   add-sqr-sqrt [<=] 98.0	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
   associate--l+ [=>] 99.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \color{blue}{\left(1 + \left(t - t\right)\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]

10. Simplified 99.8%

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

    [Start] 99.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
    +-inverses [=>] 99.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
    metadata-eval [=>] 99.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
    *-lft-identity [=>] 99.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

11. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	75.0%
Cost	65860

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := t_2 - \sqrt{z}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t_3 \leq 0.6:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{z}} + \left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{\sqrt{x} + t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_3 + \left(\left(t_4 - \sqrt{x}\right) + \left(t_1 - \sqrt{y}\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	59392

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

Alternative 3
Accuracy	97.6%
Cost	53056

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

Alternative 4
Accuracy	97.4%
Cost	52932

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 4.9 \cdot 10^{+30}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\left(t_1 - \sqrt{y}\right) + \left(t_3 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_3 + \sqrt{z}} + \left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{\sqrt{x} + t_2}\right)\\ \end{array} \]

Alternative 5
Accuracy	65.1%
Cost	40004

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 1.16 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(\left(t_2 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{z}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + t_2}\right)\\ \end{array} \]

Alternative 6
Accuracy	63.5%
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 1.16 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(\left(t_2 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \left(\sqrt{x} + \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(t_1 - \sqrt{z}\right) + \frac{1}{\sqrt{x} + t_2}\right)\\ \end{array} \]

Alternative 7
Accuracy	28.1%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} - \left(\sqrt{y} - \left(\sqrt{1 + y} - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 8
Accuracy	33.5%
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 9
Accuracy	38.4%
Cost	32904

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 10
Accuracy	38.2%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 11
Accuracy	39.0%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+15}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 12
Accuracy	39.1%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-17}:\\ \;\;\;\;t_1 + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(t_1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 13
Accuracy	39.5%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 1.06 \cdot 10^{-22}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 14
Accuracy	48.8%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 23500000000000:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15
Accuracy	48.8%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 6000000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 16
Accuracy	43.8%
Cost	13248

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

Alternative 17
Accuracy	15.4%
Cost	13120

\[\sqrt{1 + x} - \sqrt{x} \]

Alternative 18
Accuracy	34.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023160 
(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))))