Main:z from

Average Error: 8.17% → 0.38%

Time: 28.0s

Precision: binary64

Cost: 66244

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 + \sqrt{y}\\ t_4 := t_2 - \sqrt{y}\\ t_5 := \sqrt{x} + t_1\\ \mathbf{if}\;t_4 \leq 0.02:\\ \;\;\;\;\frac{t_5 + t_3 \cdot \left(1 + \left(x - x\right)\right)}{t_3 \cdot t_5}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(t_4 - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]

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))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (+ t_2 (sqrt y)))
        (t_4 (- t_2 (sqrt y)))
        (t_5 (+ (sqrt x) t_1)))
   (if (<= t_4 0.02)
     (/ (+ t_5 (* t_3 (+ 1.0 (- x x)))) (* t_3 t_5))
     (+
      (+ t_1 (- t_4 (sqrt x)))
      (+
       (/ 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) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = t_2 + sqrt(y);
	double t_4 = t_2 - sqrt(y);
	double t_5 = sqrt(x) + t_1;
	double tmp;
	if (t_4 <= 0.02) {
		tmp = (t_5 + (t_3 * (1.0 + (x - x)))) / (t_3 * t_5);
	} else {
		tmp = (t_1 + (t_4 - sqrt(x))) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	}
	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) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = t_2 + sqrt(y)
    t_4 = t_2 - sqrt(y)
    t_5 = sqrt(x) + t_1
    if (t_4 <= 0.02d0) then
        tmp = (t_5 + (t_3 * (1.0d0 + (x - x)))) / (t_3 * t_5)
    else
        tmp = (t_1 + (t_4 - sqrt(x))) + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))))
    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 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = t_2 + Math.sqrt(y);
	double t_4 = t_2 - Math.sqrt(y);
	double t_5 = Math.sqrt(x) + t_1;
	double tmp;
	if (t_4 <= 0.02) {
		tmp = (t_5 + (t_3 * (1.0 + (x - x)))) / (t_3 * t_5);
	} else {
		tmp = (t_1 + (t_4 - Math.sqrt(x))) + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))));
	}
	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 + x))
	t_2 = math.sqrt((1.0 + y))
	t_3 = t_2 + math.sqrt(y)
	t_4 = t_2 - math.sqrt(y)
	t_5 = math.sqrt(x) + t_1
	tmp = 0
	if t_4 <= 0.02:
		tmp = (t_5 + (t_3 * (1.0 + (x - x)))) / (t_3 * t_5)
	else:
		tmp = (t_1 + (t_4 - math.sqrt(x))) + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))))
	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 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_2 + sqrt(y))
	t_4 = Float64(t_2 - sqrt(y))
	t_5 = Float64(sqrt(x) + t_1)
	tmp = 0.0
	if (t_4 <= 0.02)
		tmp = Float64(Float64(t_5 + Float64(t_3 * Float64(1.0 + Float64(x - x)))) / Float64(t_3 * t_5));
	else
		tmp = Float64(Float64(t_1 + Float64(t_4 - sqrt(x))) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t)))));
	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 + x));
	t_2 = sqrt((1.0 + y));
	t_3 = t_2 + sqrt(y);
	t_4 = t_2 - sqrt(y);
	t_5 = sqrt(x) + t_1;
	tmp = 0.0;
	if (t_4 <= 0.02)
		tmp = (t_5 + (t_3 * (1.0 + (x - x)))) / (t_3 * t_5);
	else
		tmp = (t_1 + (t_4 - sqrt(x))) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (1.0 / (sqrt((1.0 + t)) + sqrt(t))));
	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 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, 0.02], N[(N[(t$95$5 + N[(t$95$3 * N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(t$95$4 - N[Sqrt[x], $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]]]]]]]

Target

Original	8.17%
Target	0.57%
Herbie	0.38%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.0200000000000000004

   1. Initial program 21.19

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

   2. Simplified 93.76

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

      [Start] 21.19	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 21.19	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      +-commutative [=>] 21.19	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+r- [=>] 21.19	\[ \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 21.19	\[ \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 21.19	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 21.19	\[ \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right)} \]
      +-commutative [=>] 21.19	\[ \sqrt{x + 1} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)\right) \]

   3. Taylor expanded in t around inf 25.32

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

   4. Taylor expanded in z around inf 25.08

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

   5. Applied egg-rr 21.28

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

   6. Applied egg-rr 15.31

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

   7. Simplified 0.95

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

      [Start] 15.31	\[ \frac{\left(1 + \left(y - y\right)\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      +-commutative [=>] 15.31	\[ \frac{\color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      +-inverses [=>] 15.31	\[ \frac{\left(\color{blue}{0} + 1\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      metadata-eval [=>] 15.31	\[ \frac{\color{blue}{1} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      distribute-lft-in [=>] 15.31	\[ \frac{\color{blue}{\left(1 \cdot \sqrt{x} + 1 \cdot \sqrt{x + 1}\right)} - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      *-lft-identity [=>] 15.31	\[ \frac{\left(1 \cdot \sqrt{x} + \color{blue}{\sqrt{x + 1}}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      *-commutative [<=] 15.31	\[ \frac{\left(\color{blue}{\sqrt{x} \cdot 1} + \sqrt{x + 1}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      +-commutative [=>] 15.31	\[ \frac{\left(\sqrt{x} \cdot 1 + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(x - \left(x + 1\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      *-commutative [=>] 15.31	\[ \frac{\left(\sqrt{x} \cdot 1 + \sqrt{1 + x}\right) - \color{blue}{\left(x - \left(x + 1\right)\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      associate--r+ [=>] 0.95	\[ \frac{\left(\sqrt{x} \cdot 1 + \sqrt{1 + x}\right) - \color{blue}{\left(\left(x - x\right) - 1\right)} \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
      *-commutative [=>] 0.95	\[ \frac{\left(\sqrt{x} \cdot 1 + \sqrt{1 + x}\right) - \left(\left(x - x\right) - 1\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}{\color{blue}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} \]

   if 0.0200000000000000004 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 2.48

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

   2. Simplified 2.47

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

      [Start] 2.48	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.47	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 2.47	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 2.47	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 2.47	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 2.47	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.47	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 2.47	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 2.47	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 2.47	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.47	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 0.76

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

   4. Simplified 0.76

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

      [Start] 0.76	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.76	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 0.76	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 0.76	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 0.76	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 0.43

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

   6. Simplified 0.13

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

      [Start] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-commutative [=>] 0.43	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\left(1 + t\right) - t\right)}\right) \]
      associate--l+ [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(1 + \left(t - t\right)\right)}\right) \]
      distribute-rgt-in [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)}\right) \]
      +-inverses [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{0} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [<=] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{\left(z - z\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      distribute-rgt-out [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(1 + \left(z - z\right)\right)}\right) \]
      +-commutative [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(\left(z - z\right) + 1\right)}\right) \]
      +-inverses [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\color{blue}{0} + 1\right)\right) \]
      metadata-eval [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 0.13	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.38

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.02:\\ \;\;\;\;\frac{\left(\sqrt{x} + \sqrt{1 + x}\right) + \left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(1 + \left(x - x\right)\right)}{\left(\sqrt{1 + y} + \sqrt{y}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.34%
Cost	92612

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

Alternative 2
Error	0.57%
Cost	53312

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

Alternative 3
Error	1.59%
Cost	53060

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 9.6 \cdot 10^{+42}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 4
Error	1.81%
Cost	52804

\[\begin{array}{l} t_1 := \sqrt{\sqrt{x} + \sqrt{1 + x}}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \frac{1}{t_2 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 - \sqrt{y}\right) + \frac{1 + \left(x - x\right)}{t_1 \cdot t_1}\\ \end{array} \]

Alternative 5
Error	2.96%
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 6
Error	4.17%
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 7
Error	3.12%
Cost	33348

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\left(2 + y \cdot 0.5\right) - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 8
Error	3.55%
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 9
Error	4.6%
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 10
Error	13.76%
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 9.8 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 11
Error	30.39%
Cost	26436

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

Alternative 12
Error	30.42%
Cost	26436

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;x \leq 36000:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 13
Error	30.61%
Cost	19908

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

Alternative 14
Error	30.49%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 15
Error	38.75%
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 38:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	30.61%
Cost	13380

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

Alternative 17
Error	35.89%
Cost	13248

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

Alternative 18
Error	64.31%
Cost	13120

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

Alternative 19
Error	65.38%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023115 
(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))))