Main:z from

Percentage Accurate: 92.0% → 99.4%

Time: 27.1s

Precision: binary64

Cost: 53312

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

Math

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

↓

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \end{array} \]

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))))

↓

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
    (/ (+ 1.0 (- y y)) (+ (sqrt y) (sqrt (+ 1.0 y)))))
   (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
  (- (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));
}

↓

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 + (y - y)) / (sqrt(y) + sqrt((1.0 + y))))) + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + (sqrt((1.0 + t)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

↓

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + ((1.0d0 + (y - y)) / (sqrt(y) + sqrt((1.0d0 + y))))) + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z)))) + (sqrt((1.0d0 + t)) - sqrt(t))
end function

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));
}

↓

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

↓

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + ((1.0 + (y - y)) / (math.sqrt(y) + math.sqrt((1.0 + y))))) + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + (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

↓

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(Float64(1.0 + Float64(y - y)) / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + 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

↓

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 + (y - y)) / (sqrt(y) + sqrt((1.0 + y))))) + (1.0 / (sqrt((1.0 + z)) + sqrt(z)))) + (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]

↓

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	92.0%
Target	99.4%
Herbie	99.4%

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

Derivation

1. Initial program 91.1%

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

2. Applied egg-rr 91.7%

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

   [Start] 91.1%	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   flip-- [=>] 91.2%	\[ \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   add-sqr-sqrt [<=] 75.2%	\[ \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 75.2%	\[ \left(\left(\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   add-sqr-sqrt [<=] 91.7%	\[ \left(\left(\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 91.7%	\[ \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

3. Simplified 92.9%

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

   [Start] 91.7%	\[ \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate--l+ [=>] 92.9%	\[ \left(\left(\frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-inverses [=>] 92.9%	\[ \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   metadata-eval [=>] 92.9%	\[ \left(\left(\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 92.9%	\[ \left(\left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied egg-rr 93.5%

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

   [Start] 92.9%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   flip-- [=>] 93.1%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   add-sqr-sqrt [<=] 71.3%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   add-sqr-sqrt [<=] 93.5%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 94.4%

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

   [Start] 93.5%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 93.5%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate--l+ [=>] 94.4%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 94.4%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 94.4%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Applied egg-rr 95.2%

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

   [Start] 94.4%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   flip-- [=>] 94.4%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   add-sqr-sqrt [<=] 78.9%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   add-sqr-sqrt [<=] 95.2%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

7. Simplified 97.0%

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

   [Start] 95.2%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\left(z + 1\right) - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [<=] 95.2%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate--l+ [=>] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{\color{blue}{1 + \left(z - z\right)}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1 + \left(z - z\right)}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Applied egg-rr 95.8%

   \[\leadsto \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - 1\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   Step-by-step derivation

   [Start] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{z + 1}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   expm1-log1p-u [=>] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{z + 1}}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   expm1-udef [=>] 95.8%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1 + \left(z - z\right)}{\sqrt{z} + \sqrt{z + 1}}\right)} - 1\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-inverses [=>] 95.8%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(e^{\mathsf{log1p}\left(\frac{1 + \color{blue}{0}}{\sqrt{z} + \sqrt{z + 1}}\right)} - 1\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   metadata-eval [=>] 95.8%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{z} + \sqrt{z + 1}}\right)} - 1\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 95.8%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right)} - 1\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

9. Simplified 97.0%

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

   [Start] 95.8%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - 1\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   expm1-def [=>] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   expm1-log1p [=>] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   +-commutative [=>] 97.0%	\[ \left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

10. Final simplification 97.0%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	53312

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

Alternative 2
Accuracy	99.2%
Cost	92484

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + z}\\ t_5 := t_4 - \sqrt{z}\\ t_6 := \sqrt{y} + t_3\\ \mathbf{if}\;\left(\left(t_1 - \sqrt{x}\right) + \left(t_3 - \sqrt{y}\right)\right) + t_5 \leq 0.8:\\ \;\;\;\;t_2 + \left(t_5 + \left(\frac{1}{t_1 + \sqrt{x}} + \frac{1}{t_6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\left(\frac{1 + \left(y - y\right)}{t_6} + \frac{1}{\left(1 + x \cdot 0.5\right) + \sqrt{x}}\right) + \frac{1 + \left(z - z\right)}{t_4 + \sqrt{z}}\right)\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	65988

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_2 - \sqrt{x} \leq 0.8:\\ \;\;\;\;t_4 + \left(\left(t_1 - \sqrt{z}\right) + \left(\frac{1}{t_2 + \sqrt{x}} + \left(t_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 + \left(\left(\frac{1 + \left(y - y\right)}{\sqrt{y} + t_3} + \frac{1}{\left(1 + x \cdot 0.5\right) + \sqrt{x}}\right) + \frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}}\right)\\ \end{array} \]

Alternative 4
Accuracy	98.4%
Cost	47561

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.22 \cdot 10^{-14} \lor \neg \left(y \leq 1.15 \cdot 10^{+31}\right):\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} + \left(\frac{1}{t_2 + \sqrt{x}} + \frac{1 + \left(y - y\right)}{\sqrt{y} + \left(1 + y \cdot 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	98.5%
Cost	47428

\[\begin{array}{l} t_1 := 1 + \left(y - y\right)\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\\ \mathbf{if}\;y \leq 5 \cdot 10^{+32}:\\ \;\;\;\;t_2 + \left(\left(\frac{t_1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\left(1 + x \cdot 0.5\right) + \sqrt{x}}\right) + t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(t_3 + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{t_1}{\sqrt{y} + \left(1 + y \cdot 0.5\right)}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	98.2%
Cost	47176

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t_1 - \sqrt{z}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \frac{1}{t_3 + \sqrt{x}}\\ \mathbf{if}\;y \leq 3 \cdot 10^{-30}:\\ \;\;\;\;t_4 + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} + \left(1 + t_5\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;t_3 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(t_2 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 + \left(t_2 + \left(t_5 + \frac{1 + \left(y - y\right)}{\sqrt{y} + \left(1 + y \cdot 0.5\right)}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	97.6%
Cost	40260

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \frac{1}{t_1 + \sqrt{x}}\\ t_3 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\left(t_3 + \left(1 + t_2\right)\right) + \frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(t_3 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	98.2%
Cost	40260

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \frac{1}{t_2 + \sqrt{x}}\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} + \left(1 + t_3\right)\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Accuracy	97.2%
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 3.4 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 + 2\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;t_2 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}}\\ \end{array} \]

Alternative 10
Accuracy	96.5%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.05 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+31}:\\ \;\;\;\;\left(t_1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 11
Accuracy	90.1%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 12
Accuracy	90.1%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(t_1 + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 13
Accuracy	95.4%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\left(t_1 + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 14
Accuracy	90.0%
Cost	20296

\[\begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 15
Accuracy	89.8%
Cost	19716

\[\begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\mathsf{hypot}\left(1, \sqrt{z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 16
Accuracy	89.8%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 17
Accuracy	85.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 18
Accuracy	85.0%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 19
Accuracy	61.9%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 0.92:\\ \;\;\;\;\left(x \cdot 0.5 + 2\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 20
Accuracy	64.8%
Cost	13248

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

Alternative 21
Accuracy	61.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 0.92:\\ \;\;\;\;\left(x \cdot 0.5 + 2\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]

Alternative 22
Accuracy	43.7%
Cost	6848

\[\left(x \cdot 0.5 + 2\right) - \sqrt{x} \]

Alternative 23
Accuracy	4.5%
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023167 
(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))))