Main:z from

Average Error: 5.5 → 0.5

Time: 27.5s

Precision: binary64

Cost: 66116

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x}\\ t_3 := t_2 - \sqrt{x}\\ t_4 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{if}\;t_3 \leq 0.999:\\ \;\;\;\;t_4 + \left(t_1 + \frac{1}{\sqrt{x} + t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(t_4 + \left(t_1 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right)\\ \end{array} \]

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 t)) (sqrt t)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- t_2 (sqrt x)))
        (t_4 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))
   (if (<= t_3 0.999)
     (+ t_4 (+ t_1 (/ 1.0 (+ (sqrt x) t_2))))
     (+ t_3 (+ t_4 (+ t_1 (/ 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((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + x));
	double t_3 = t_2 - sqrt(x);
	double t_4 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	double tmp;
	if (t_3 <= 0.999) {
		tmp = t_4 + (t_1 + (1.0 / (sqrt(x) + t_2)));
	} else {
		tmp = t_3 + (t_4 + (t_1 + (1.0 / (sqrt((1.0 + z)) + 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 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + x))
    t_3 = t_2 - sqrt(x)
    t_4 = 1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))
    if (t_3 <= 0.999d0) then
        tmp = t_4 + (t_1 + (1.0d0 / (sqrt(x) + t_2)))
    else
        tmp = t_3 + (t_4 + (t_1 + (1.0d0 / (sqrt((1.0d0 + z)) + 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 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = t_2 - Math.sqrt(x);
	double t_4 = 1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y));
	double tmp;
	if (t_3 <= 0.999) {
		tmp = t_4 + (t_1 + (1.0 / (Math.sqrt(x) + t_2)));
	} else {
		tmp = t_3 + (t_4 + (t_1 + (1.0 / (Math.sqrt((1.0 + z)) + 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 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + x))
	t_3 = t_2 - math.sqrt(x)
	t_4 = 1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))
	tmp = 0
	if t_3 <= 0.999:
		tmp = t_4 + (t_1 + (1.0 / (math.sqrt(x) + t_2)))
	else:
		tmp = t_3 + (t_4 + (t_1 + (1.0 / (math.sqrt((1.0 + z)) + 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 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(t_2 - sqrt(x))
	t_4 = Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))
	tmp = 0.0
	if (t_3 <= 0.999)
		tmp = Float64(t_4 + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + t_2))));
	else
		tmp = Float64(t_3 + Float64(t_4 + Float64(t_1 + Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + 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 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + x));
	t_3 = t_2 - sqrt(x);
	t_4 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	tmp = 0.0;
	if (t_3 <= 0.999)
		tmp = t_4 + (t_1 + (1.0 / (sqrt(x) + t_2)));
	else
		tmp = t_3 + (t_4 + (t_1 + (1.0 / (sqrt((1.0 + z)) + 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[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.999], N[(t$95$4 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$4 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.5
Target	0.4
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.998999999999999999

   1. Initial program 50.6

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

   2. Simplified 50.6

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

      [Start] 50.6	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 50.6	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 50.6	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+r+ [<=] 50.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 50.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 50.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 50.6	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Applied egg-rr 8.6

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

   4. Simplified 8.6

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

      [Start] 8.6	\[ \left(1 + \left(x - x\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 8.6	\[ \color{blue}{\left(\left(x - x\right) + 1\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 8.6	\[ \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 8.6	\[ \color{blue}{1} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 8.6	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 8.6	\[ \frac{1}{\color{blue}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   5. Applied egg-rr 1.3

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

   6. Simplified 1.3

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

      [Start] 1.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 1.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 1.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 1.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 1.3	\[ \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   7. Applied egg-rr 42.2

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

   8. Simplified 1.3

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

      [Start] 42.2	\[ \left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} - 1\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      expm1-def [=>] 1.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right)} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      expm1-log1p [=>] 1.3	\[ \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   9. Taylor expanded in z around inf 61.9

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

   10. Simplified 1.3

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

       [Start] 61.9	\[ \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \sqrt{1 + t}\right)\right) - \sqrt{t} \]
       associate--l+ [=>] 58.1	\[ \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
       +-commutative [=>] 58.1	\[ \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]
       associate--l+ [=>] 1.3	\[ \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]

   if 0.998999999999999999 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 2.1

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

   2. Simplified 2.1

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

      [Start] 2.1	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.1	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.1	\[ \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+r+ [<=] 2.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right)\right) \]

   3. Applied egg-rr 1.3

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

   4. Simplified 1.3

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

      [Start] 1.3	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 1.3	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 1.3	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 1.3	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 1.3	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

   5. Applied egg-rr 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-commutative [=>] 0.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      +-inverses [=>] 0.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      metadata-eval [=>] 0.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      *-lft-identity [=>] 0.4	\[ \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.999:\\ \;\;\;\;\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	65988

\[\begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 - \sqrt{y}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t_3 \leq 10^{-5}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(\left(t_1 - \sqrt{t}\right) + \frac{1}{\sqrt{x} + t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_4 - \sqrt{x}\right) + \left(t_3 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{t_1 + \sqrt{t}}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.7
Cost	65988

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 - \sqrt{y}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t_3 \leq 10^{-5}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(t_1 + \frac{1}{\sqrt{x} + t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_4 - \sqrt{x}\right) + \left(\left(t_1 + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right) + t_3\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	65860

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 - \sqrt{y}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t_3 \leq 10^{-5}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(t_1 + \frac{1}{\sqrt{x} + t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(t_4 - \sqrt{x}\right) + t_3\right)\right)\\ \end{array} \]

Alternative 4
Error	1.5
Cost	65860

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 - \sqrt{y}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t_3 \leq 10^{-5}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(t_1 + \frac{1}{\sqrt{x} + t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_4 - \sqrt{x}\right) + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_1\right) + t_3\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	52928

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

Alternative 6
Error	1.3
Cost	39876

\[\begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 3.4 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + t_1\right) + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{y}} + \left(t_1 + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]

Alternative 7
Error	2.8
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 8
Error	5.5
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{1 + t} + \left(1 + t_1\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 9
Error	6.4
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 6 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 10
Error	6.2
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 11
Error	6.6
Cost	19908

\[\begin{array}{l} \mathbf{if}\;y \leq 7.4 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 12
Error	19.2
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13
Error	22.8
Cost	13248

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

Alternative 14
Error	40.9
Cost	13120

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

Alternative 15
Error	41.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(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))))