Main:z from

Average Accuracy: 91.2% → 99.9%

Time: 28.7s

Precision: binary64

Cost: 59520

Math

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

↓

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

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 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(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(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[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	91.2%
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 91.2%

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

2. Simplified 91.2%

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

   [Start] 91.2	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 91.2	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 91.2	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
   associate-+r+ [<=] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
   +-commutative [=>] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
   sub-neg [=>] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{t + 1} + \left(-\sqrt{t}\right)\right)}\right)\right) \]
   +-commutative [=>] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} + \left(-\sqrt{t}\right)\right)\right)\right) \]
   sub-neg [<=] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)}\right)\right) \]
   +-commutative [=>] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

3. Applied egg-rr 93.2%

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

   [Start] 91.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   flip-- [=>] 91.3	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   div-inv [=>] 91.3	\[ \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [<=] 72.0	\[ \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   +-commutative [=>] 72.0	\[ \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [<=] 91.6	\[ \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   associate--l+ [=>] 93.2	\[ \color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   +-commutative [=>] 93.2	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [=>] 93.2	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   hypot-1-def [=>] 93.2	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

4. Simplified 93.2%

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

   [Start] 93.2	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   +-inverses [=>] 93.2	\[ \left(1 + \color{blue}{0}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   metadata-eval [=>] 93.2	\[ \color{blue}{1} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   *-lft-identity [=>] 93.2	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   +-commutative [=>] 93.2	\[ \frac{1}{\color{blue}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

5. Applied egg-rr 93.6%

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

   [Start] 93.2	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   flip-- [=>] 93.4	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\frac{\sqrt{1 + y} \cdot \sqrt{1 + y} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [<=] 72.7	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [<=] 93.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

6. Simplified 95.3%

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

   [Start] 93.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   associate--l+ [=>] 95.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   +-inverses [=>] 95.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   metadata-eval [=>] 95.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{1}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

7. Applied egg-rr 97.6%

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

   [Start] 95.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   flip-- [=>] 95.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   div-inv [=>] 95.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [<=] 75.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   add-sqr-sqrt [<=] 95.8	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   associate--l+ [=>] 97.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

8. Simplified 97.6%

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

   [Start] 97.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   +-inverses [=>] 97.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   metadata-eval [=>] 97.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   *-lft-identity [=>] 97.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

9. Applied egg-rr 99.9%

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

   [Start] 97.6	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
   flip-- [=>] 97.7	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{1 + t} + \sqrt{t}}}\right)\right) \]
   div-inv [=>] 97.7	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\sqrt{1 + t} \cdot \sqrt{1 + t} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right)\right) \]
   add-sqr-sqrt [<=] 76.4	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{\left(1 + t\right)} - \sqrt{t} \cdot \sqrt{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
   add-sqr-sqrt [<=] 98.0	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\right) - \color{blue}{t}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
   associate--l+ [=>] 99.9	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(1 + \left(t - t\right)\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]

10. Simplified 99.9%

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

    [Start] 99.9	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
    +-inverses [=>] 99.9	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
    metadata-eval [=>] 99.9	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
    *-lft-identity [=>] 99.9	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right)\right) \]

11. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	52.8%
Cost	65988

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;t_2 - \sqrt{z} \leq 0.005:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{t_2 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t_2 + \left(\left(t_1 - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)\right) + \sqrt{1 + x}\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	59392

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

Alternative 3
Accuracy	54.5%
Cost	46340

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(2 + \left(t_1 + \sqrt{1 + t}\right)\right) + \left(\frac{t - z}{\sqrt{z} - \sqrt{t}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{t_1 + \sqrt{z}}\right)\\ \end{array} \]

Alternative 4
Accuracy	33.8%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(2 + \left(t_1 + \sqrt{1 + t}\right)\right) + \left(\frac{t - z}{\sqrt{z} - \sqrt{t}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{t_1 + \sqrt{z}} + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	55.8%
Cost	39744

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

Alternative 6
Accuracy	33.6%
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(2 + \left(t_1 + \sqrt{1 + t}\right)\right) + \left(\frac{t - z}{\sqrt{z} - \sqrt{t}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_1 - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{x}\right) - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	32.9%
Cost	33348

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(2 + \left(t_1 + \sqrt{1 + t}\right)\right) + \left(\frac{t - z}{\sqrt{z} - \sqrt{t}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_1 - \sqrt{z}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	35.4%
Cost	33092

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;\left(2 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_1 - \sqrt{z}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	54.3%
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.02 \cdot 10^{+17}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\sqrt{y} - t_1\right)\\ \end{array} \]

Alternative 10
Accuracy	38.9%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Accuracy	49.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-28}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\sqrt{y} - \sqrt{1 + y}\right)\\ \end{array} \]

Alternative 12
Accuracy	44.5%
Cost	13248

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

Alternative 13
Accuracy	34.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023147 
(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))))