Main:z from

Average Error: 5.5 → 0.1

Time: 26.3s

Precision: binary64

Cost: 66116

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

Math

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

↓

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

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 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- t_2 (sqrt y)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= t_3 0.5)
     (+ (/ 1.0 (+ t_2 (sqrt y))) (+ t_1 (/ 1.0 (+ (sqrt x) t_4))))
     (+
      (- t_4 (sqrt x))
      (+ t_3 (+ t_1 (/ 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 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	double t_2 = sqrt((1.0 + y));
	double t_3 = t_2 - sqrt(y);
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 0.5) {
		tmp = (1.0 / (t_2 + sqrt(y))) + (t_1 + (1.0 / (sqrt(x) + t_4)));
	} else {
		tmp = (t_4 - sqrt(x)) + (t_3 + (t_1 + (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) :: tmp
    t_1 = 1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))
    t_2 = sqrt((1.0d0 + y))
    t_3 = t_2 - sqrt(y)
    t_4 = sqrt((1.0d0 + x))
    if (t_3 <= 0.5d0) then
        tmp = (1.0d0 / (t_2 + sqrt(y))) + (t_1 + (1.0d0 / (sqrt(x) + t_4)))
    else
        tmp = (t_4 - sqrt(x)) + (t_3 + (t_1 + (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 = 1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = t_2 - Math.sqrt(y);
	double t_4 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 0.5) {
		tmp = (1.0 / (t_2 + Math.sqrt(y))) + (t_1 + (1.0 / (Math.sqrt(x) + t_4)));
	} else {
		tmp = (t_4 - Math.sqrt(x)) + (t_3 + (t_1 + (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 = 1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))
	t_2 = math.sqrt((1.0 + y))
	t_3 = t_2 - math.sqrt(y)
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_3 <= 0.5:
		tmp = (1.0 / (t_2 + math.sqrt(y))) + (t_1 + (1.0 / (math.sqrt(x) + t_4)))
	else:
		tmp = (t_4 - math.sqrt(x)) + (t_3 + (t_1 + (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 = Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_2 - sqrt(y))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_3 <= 0.5)
		tmp = Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + t_4))));
	else
		tmp = Float64(Float64(t_4 - sqrt(x)) + Float64(t_3 + Float64(t_1 + 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 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	t_2 = sqrt((1.0 + y));
	t_3 = t_2 - sqrt(y);
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_3 <= 0.5)
		tmp = (1.0 / (t_2 + sqrt(y))) + (t_1 + (1.0 / (sqrt(x) + t_4)));
	else
		tmp = (t_4 - sqrt(x)) + (t_3 + (t_1 + (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[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.5], N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.5
Target	0.4
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.5

   1. Initial program 13.8

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

   2. Simplified 13.8

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

      [Start] 13.8	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 13.8	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 13.8	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+r+ [<=] 13.8	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 13.8	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 13.8	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 13.8	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Applied egg-rr 4.9

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

   4. Simplified 4.9

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

      [Start] 4.9	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 4.9	\[ \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 4.9	\[ \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 4.9	\[ \color{blue}{1} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 4.9	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 4.9	\[ \frac{1}{\color{blue}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   5. Applied egg-rr 3.5

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

   6. Simplified 0.7

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

      [Start] 3.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(y + \left(1 - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate-*r/ [=>] 3.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\frac{\left(y + \left(1 - y\right)\right) \cdot 1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-rgt-identity [=>] 3.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{y + \left(1 - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate-+r- [=>] 3.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [<=] 3.5	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      associate--l+ [=>] 0.7	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{\color{blue}{1 + \left(y - y\right)}}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   7. Applied egg-rr 0.1

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

   8. Simplified 0.1

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

      [Start] 0.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 0.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 0.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 0.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 0.1	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   9. Taylor expanded in t around inf 0.2

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

   10. Simplified 0.2

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

       [Start] 0.2	\[ \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) \]
       +-commutative [=>] 0.2	\[ \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]

   if 0.5 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 1.6

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

   2. Simplified 1.6

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

      [Start] 1.6	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 1.6	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 1.6	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+r+ [<=] 1.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 1.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 1.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 1.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Applied egg-rr 1.2

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

   4. Simplified 1.2

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

      [Start] 1.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(t - t\right)\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-commutative [=>] 1.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\left(\left(t - t\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [=>] 1.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      metadata-eval [=>] 1.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{1} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      *-lft-identity [=>] 1.2	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right)\right) \]

   5. Applied egg-rr 0.1

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

   6. Simplified 0.1

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

      [Start] 0.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-commutative [=>] 0.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [=>] 0.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      metadata-eval [=>] 0.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      *-lft-identity [=>] 0.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.5:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	59648

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

Alternative 2
Error	0.2
Cost	52932

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 0.122:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(t_1 - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t_3 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{y}} + \left(\frac{1}{t_3 + \sqrt{z}} + \frac{1}{\sqrt{x} + t_2}\right)\\ \end{array} \]

Alternative 3
Error	2.0
Cost	40008

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-66}:\\ \;\;\;\;\left(t_3 - \sqrt{x}\right) + \left(1 + \left(\left(t_1 - \sqrt{t}\right) + \left(t_2 - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+31}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{t_2 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_3} + \left(\left(1 + \left(t_2 + t_1\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	40004

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 1.26 \cdot 10^{+25}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(t_1 - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{x} + t_2}\right)\\ \end{array} \]

Alternative 5
Error	1.4
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 2.25 \cdot 10^{-35}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(t_1 - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + t_2}\right)\\ \end{array} \]

Alternative 6
Error	2.0
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-66}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 - \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+31}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{t_1 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \]

Alternative 7
Error	5.0
Cost	26820

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+31}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 8
Error	5.8
Cost	26692

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+31}:\\ \;\;\;\;1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 9
Error	6.5
Cost	26564

\[\begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 10
Error	6.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 2.75 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 11
Error	10.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 12
Error	10.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 13
Error	22.9
Cost	13248

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

Alternative 14
Error	41.1
Cost	13120

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

Alternative 15
Error	41.5
Cost	6848

\[\left(1 + x \cdot 0.5\right) - \sqrt{x} \]

Alternative 16
Error	41.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(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))))