Main:z from

Average Error: 8.62% → 0.37%

Time: 24.8s

Precision: binary64

Cost: 53060

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 1.35 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{1}{t_4 + \sqrt{t}} + \frac{1}{t_1 + \sqrt{z}}\right) + \left(t_2 + \left(\left(t_3 - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_3 + \sqrt{y}} + \frac{1}{t_2 + \sqrt{x}}\right) + \left(\left(t_4 - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (sqrt (+ 1.0 t))))
   (if (<= z 1.35e+28)
     (+
      (+ (/ 1.0 (+ t_4 (sqrt t))) (/ 1.0 (+ t_1 (sqrt z))))
      (+ t_2 (- (- t_3 (sqrt y)) (sqrt x))))
     (+
      (+ (/ 1.0 (+ t_3 (sqrt y))) (/ 1.0 (+ t_2 (sqrt x))))
      (+ (- t_4 (sqrt t)) (- t_1 (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((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((1.0 + t));
	double tmp;
	if (z <= 1.35e+28) {
		tmp = ((1.0 / (t_4 + sqrt(t))) + (1.0 / (t_1 + sqrt(z)))) + (t_2 + ((t_3 - sqrt(y)) - sqrt(x)));
	} else {
		tmp = ((1.0 / (t_3 + sqrt(y))) + (1.0 / (t_2 + sqrt(x)))) + ((t_4 - sqrt(t)) + (t_1 - sqrt(z)));
	}
	return tmp;
}

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
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((1.0d0 + y))
    t_4 = sqrt((1.0d0 + t))
    if (z <= 1.35d+28) then
        tmp = ((1.0d0 / (t_4 + sqrt(t))) + (1.0d0 / (t_1 + sqrt(z)))) + (t_2 + ((t_3 - sqrt(y)) - sqrt(x)))
    else
        tmp = ((1.0d0 / (t_3 + sqrt(y))) + (1.0d0 / (t_2 + sqrt(x)))) + ((t_4 - sqrt(t)) + (t_1 - sqrt(z)))
    end if
    code = tmp
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((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((1.0 + y));
	double t_4 = Math.sqrt((1.0 + t));
	double tmp;
	if (z <= 1.35e+28) {
		tmp = ((1.0 / (t_4 + Math.sqrt(t))) + (1.0 / (t_1 + Math.sqrt(z)))) + (t_2 + ((t_3 - Math.sqrt(y)) - Math.sqrt(x)));
	} else {
		tmp = ((1.0 / (t_3 + Math.sqrt(y))) + (1.0 / (t_2 + Math.sqrt(x)))) + ((t_4 - Math.sqrt(t)) + (t_1 - Math.sqrt(z)));
	}
	return tmp;
}

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((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((1.0 + y))
	t_4 = math.sqrt((1.0 + t))
	tmp = 0
	if z <= 1.35e+28:
		tmp = ((1.0 / (t_4 + math.sqrt(t))) + (1.0 / (t_1 + math.sqrt(z)))) + (t_2 + ((t_3 - math.sqrt(y)) - math.sqrt(x)))
	else:
		tmp = ((1.0 / (t_3 + math.sqrt(y))) + (1.0 / (t_2 + math.sqrt(x)))) + ((t_4 - math.sqrt(t)) + (t_1 - math.sqrt(z)))
	return tmp

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 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (z <= 1.35e+28)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(t))) + Float64(1.0 / Float64(t_1 + sqrt(z)))) + Float64(t_2 + Float64(Float64(t_3 - sqrt(y)) - sqrt(x))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(y))) + Float64(1.0 / Float64(t_2 + sqrt(x)))) + Float64(Float64(t_4 - sqrt(t)) + Float64(t_1 - sqrt(z))));
	end
	return tmp
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_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((1.0 + y));
	t_4 = sqrt((1.0 + t));
	tmp = 0.0;
	if (z <= 1.35e+28)
		tmp = ((1.0 / (t_4 + sqrt(t))) + (1.0 / (t_1 + sqrt(z)))) + (t_2 + ((t_3 - sqrt(y)) - sqrt(x)));
	else
		tmp = ((1.0 / (t_3 + sqrt(y))) + (1.0 / (t_2 + sqrt(x)))) + ((t_4 - sqrt(t)) + (t_1 - sqrt(z)));
	end
	tmp_2 = tmp;
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[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.35e+28], N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	8.62%
Target	0.64%
Herbie	0.37%

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

Derivation

1. Split input into 2 regimes

2. if z < 1.3500000000000001e28

   1. Initial program 4.78

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

   2. Simplified 4.78

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

      [Start] 4.78	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 4.78	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 4.78	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 4.8	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 4.8	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 4.78	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 4.78	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 4.78	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 4.78	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 4.78	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 4.78	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 1.72

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

   4. Simplified 1.72

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

      [Start] 1.72	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 1.72	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 1.72	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 1.72	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 1.72	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 0.43

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

   6. Simplified 0.43

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

      [Start] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-commutative [=>] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(\left(t - t\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      +-inverses [=>] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      metadata-eval [=>] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-lft-identity [=>] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

   if 1.3500000000000001e28 < z

   1. Initial program 11.02

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

   2. Simplified 11.02

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

      [Start] 11.02	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 11.02	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 11.02	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 13.14	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 13.14	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 11.02	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 11.02	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 11.02	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 11.02	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 11.02	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 11.02	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 11.02

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

   4. Applied egg-rr 3.73

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

   5. Simplified 3.73

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

      [Start] 3.73	\[ e^{\log \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 3.73	\[ e^{\log \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 3.73	\[ e^{\log \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 3.73	\[ e^{\log \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 3.73	\[ e^{\log \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   6. Applied egg-rr 0.83

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

   7. Simplified 0.83

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

      [Start] 0.83	\[ e^{\log \left(\left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.83	\[ e^{\log \left(\color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 0.83	\[ e^{\log \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 0.83	\[ e^{\log \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 0.83	\[ e^{\log \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   8. Applied egg-rr 0.33

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.37

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.97%
Cost	65856

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

Alternative 2
Error	2.31%
Cost	65540

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_2 - \sqrt{x} \leq 0.99999996:\\ \;\;\;\;\left(t_3 + \left(t_1 - \sqrt{z}\right)\right) + e^{-\log \left(t_2 + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t_3 + \frac{1}{t_1 + \sqrt{z}}\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \end{array} \]

Alternative 3
Error	0.41%
Cost	53060

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{1 + t}\\ \mathbf{if}\;z \leq 1.35 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{1}{t_3 + \sqrt{t}} + \frac{1}{t_1 + \sqrt{z}}\right) + \left(1 + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_2 + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) + \left(\left(t_3 - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 4
Error	8.32%
Cost	39872

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

Alternative 5
Error	7.73%
Cost	39872

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

Alternative 6
Error	7.77%
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 0.00155:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right) + \left(1 + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(1 + \frac{1}{t_1 + \sqrt{z}}\right)\\ \end{array} \]

Alternative 7
Error	8.81%
Cost	39744

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

Alternative 8
Error	7.91%
Cost	26820

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

Alternative 9
Error	8.99%
Cost	26692

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \frac{1}{t_1 + \sqrt{z}}\right)\\ \end{array} \]

Alternative 10
Error	14.04%
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 11
Error	9.48%
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 12
Error	14.32%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;z \leq 56000000000000:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(2 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 13
Error	15.7%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;z \leq 47000000000000:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(2 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 14
Error	16.24%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15
Error	36.21%
Cost	13248

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

Alternative 16
Error	64.65%
Cost	13120

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

Alternative 17
Error	65.87%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023102 
(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))))