Main:z from

Average Accuracy: 91.5% → 99.3%

Time: 28.2s

Precision: binary64

Cost: 79936

\[ \begin{array}{c}[x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{x} + \sqrt{1 + x}\\ t_2 := \sqrt{1 + y} + \sqrt{y}\\ \frac{\left(1 + \left(x - x\right)\right) \cdot t_2 + t_1 \cdot \left(1 + \left(y - y\right)\right)}{t_2 \cdot t_1} + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) \end{array} \]

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))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt x) (sqrt (+ 1.0 x))))
        (t_2 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (+
    (/ (+ (* (+ 1.0 (- x x)) t_2) (* t_1 (+ 1.0 (- y y)))) (* t_2 t_1))
    (+ (- (sqrt (+ 1.0 t)) (sqrt t)) (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))))))

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) {
	double t_1 = sqrt(x) + sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y)) + sqrt(y);
	return ((((1.0 + (x - x)) * t_2) + (t_1 * (1.0 + (y - y)))) / (t_2 * t_1)) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (sqrt((1.0 + z)) + sqrt(z))));
}

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

↓

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
    real(8) :: t_1
    real(8) :: t_2
    t_1 = sqrt(x) + sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y)) + sqrt(y)
    code = ((((1.0d0 + (x - x)) * t_2) + (t_1 * (1.0d0 + (y - y)))) / (t_2 * t_1)) + ((sqrt((1.0d0 + t)) - sqrt(t)) + (1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))))
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));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(x) + Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y)) + Math.sqrt(y);
	return ((((1.0 + (x - x)) * t_2) + (t_1 * (1.0 + (y - y)))) / (t_2 * t_1)) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) + (1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))));
}

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):
	t_1 = math.sqrt(x) + math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y)) + math.sqrt(y)
	return ((((1.0 + (x - x)) * t_2) + (t_1 * (1.0 + (y - y)))) / (t_2 * t_1)) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) + (1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))))

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)
	t_1 = Float64(sqrt(x) + sqrt(Float64(1.0 + x)))
	t_2 = Float64(sqrt(Float64(1.0 + y)) + sqrt(y))
	return Float64(Float64(Float64(Float64(Float64(1.0 + Float64(x - x)) * t_2) + Float64(t_1 * Float64(1.0 + Float64(y - y)))) / Float64(t_2 * t_1)) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))))
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)
	t_1 = sqrt(x) + sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y)) + sqrt(y);
	tmp = ((((1.0 + (x - x)) * t_2) + (t_1 * (1.0 + (y - y)))) / (t_2 * t_1)) + ((sqrt((1.0 + t)) - sqrt(t)) + (1.0 / (sqrt((1.0 + z)) + sqrt(z))));
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_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	91.5%
Target	99.4%
Herbie	99.3%

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

Derivation

1. Initial program 91.5%

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

2. Simplified 91.5%

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

   [Start] 91.5	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   associate-+l+ [=>] 91.5	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
   associate-+l- [=>] 91.5	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   associate--r- [=>] 90.3	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   remove-double-neg [<=] 90.3	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   associate-+l- [=>] 91.5	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 91.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   remove-double-neg [=>] 91.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [=>] 91.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   sub-neg [<=] 91.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 91.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

3. Applied egg-rr 92.9%

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

   [Start] 91.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   flip-- [=>] 91.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   div-inv [=>] 91.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 65.2	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   add-sqr-sqrt [<=] 92.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\left(1 + z\right) - \color{blue}{z}\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--l+ [=>] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(1 + \left(z - z\right)\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

4. Simplified 92.9%

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

   [Start] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-inverses [=>] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   *-lft-identity [=>] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

5. Applied egg-rr 92.9%

   \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   Proof

   [Start] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   expm1-log1p-u [=>] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

6. Applied egg-rr 92.9%

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

   [Start] 92.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate--r- [=>] 92.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   expm1-udef [=>] 92.9	\[ \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)} - 1\right)}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   sub-neg [=>] 92.9	\[ \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)} + \left(-1\right)\right)}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-+r+ [=>] 92.8	\[ \color{blue}{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + e^{\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)}\right) + \left(-1\right)\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

7. Applied egg-rr 99.3%

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

   [Start] 92.9	\[ \left(\left(\left(1 + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + -1\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   expm1-log1p-u [=>] 91.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(1 + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + -1\right)\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   log1p-udef [=>] 92.8	\[ \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\left(\left(1 + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + -1\right)\right)}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   +-commutative [=>] 92.8	\[ \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(-1 + \left(\left(1 + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-+l+ [=>] 92.8	\[ \mathsf{expm1}\left(\log \left(1 + \left(-1 + \color{blue}{\left(1 + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-+r+ [=>] 92.8	\[ \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(\left(-1 + 1\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 92.8	\[ \mathsf{expm1}\left(\log \left(1 + \left(\color{blue}{0} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [<=] 92.8	\[ \mathsf{expm1}\left(\log \left(1 + \left(\color{blue}{\log 1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-+r+ [=>] 92.8	\[ \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + \log 1\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 92.8	\[ \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{0}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   metadata-eval [=>] 92.8	\[ \mathsf{expm1}\left(\log \left(\color{blue}{1} + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   associate-+l+ [<=] 92.8	\[ \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

8. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	79168

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{x} + \sqrt{1 + x}\\ \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \frac{t_1 + \left(\sqrt{y} + t_2\right)}{\left(t_1 + \sqrt{y}\right) \cdot t_2} \end{array} \]

Alternative 2
Accuracy	96.9%
Cost	65856

\[\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + {\left({\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)}^{3}\right)}^{0.3333333333333333} \]

Alternative 3
Accuracy	92.7%
Cost	52936

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t_1 - \sqrt{z}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{t_4 + \sqrt{t}} + t_2\right) + \left(\left(1 + t_3\right) - \sqrt{y}\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+72}:\\ \;\;\;\;1 + \left(t_3 + \left(\frac{1}{t_1 + \sqrt{z}} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_4 - \sqrt{t}\right) + \left(t_2 + \left(\left(t_3 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	93.4%
Cost	52928

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

Alternative 5
Accuracy	92.1%
Cost	52800

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

Alternative 6
Accuracy	92.9%
Cost	52800

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

Alternative 7
Accuracy	91.8%
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 0.0031:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(1 + t_1\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	92.4%
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 0.0031:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\left(1 + t_1\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 9
Accuracy	91.5%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	90.2%
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + z \cdot 0.5\right) - \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	91.4%
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(1 + z \cdot 0.5\right) - \sqrt{z}\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 12
Accuracy	91.1%
Cost	26564

\[\begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;2 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 13
Accuracy	84.0%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+26}:\\ \;\;\;\;\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	84.0%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 10^{+26}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	84.9%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + 2\\ \mathbf{elif}\;y \leq 10^{+26}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Accuracy	80.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.95:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	52.0%
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 0.118:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Accuracy	34.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023135 
(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))))