Main:z from

Average Error: 5.1 → 0.2

Time: 28.7s

Precision: binary64

Cost: 79044

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z} + \sqrt{z}\\ t_3 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(t_3 + \frac{1}{t_1 + \sqrt{x}}\right)\right) + \left(\frac{1}{t_2} + \left(t_4 - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(t_3 - \sqrt{x}\right)\right) + \left(\frac{1}{t_4 + \sqrt{t}} - \frac{-1}{t_2}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (+ (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
        (t_4 (sqrt (+ 1.0 t))))
   (if (<= (- t_1 (sqrt x)) 5e-5)
     (+
      (expm1 (log1p (+ t_3 (/ 1.0 (+ t_1 (sqrt x))))))
      (+ (/ 1.0 t_2) (- t_4 (sqrt t))))
     (+ (+ t_1 (- t_3 (sqrt x))) (- (/ 1.0 (+ t_4 (sqrt t))) (/ -1.0 t_2))))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + z)) + sqrt(z);
	double t_3 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	double t_4 = sqrt((1.0 + t));
	double tmp;
	if ((t_1 - sqrt(x)) <= 5e-5) {
		tmp = expm1(log1p((t_3 + (1.0 / (t_1 + sqrt(x)))))) + ((1.0 / t_2) + (t_4 - sqrt(t)));
	} else {
		tmp = (t_1 + (t_3 - sqrt(x))) + ((1.0 / (t_4 + sqrt(t))) - (-1.0 / t_2));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z)) + Math.sqrt(z);
	double t_3 = 1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y));
	double t_4 = Math.sqrt((1.0 + t));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 5e-5) {
		tmp = Math.expm1(Math.log1p((t_3 + (1.0 / (t_1 + Math.sqrt(x)))))) + ((1.0 / t_2) + (t_4 - Math.sqrt(t)));
	} else {
		tmp = (t_1 + (t_3 - Math.sqrt(x))) + ((1.0 / (t_4 + Math.sqrt(t))) - (-1.0 / t_2));
	}
	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((x + 1.0))
	t_2 = math.sqrt((1.0 + z)) + math.sqrt(z)
	t_3 = 1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))
	t_4 = math.sqrt((1.0 + t))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 5e-5:
		tmp = math.expm1(math.log1p((t_3 + (1.0 / (t_1 + math.sqrt(x)))))) + ((1.0 / t_2) + (t_4 - math.sqrt(t)))
	else:
		tmp = (t_1 + (t_3 - math.sqrt(x))) + ((1.0 / (t_4 + math.sqrt(t))) - (-1.0 / t_2))
	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(x + 1.0))
	t_2 = Float64(sqrt(Float64(1.0 + z)) + sqrt(z))
	t_3 = Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))
	t_4 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 5e-5)
		tmp = Float64(expm1(log1p(Float64(t_3 + Float64(1.0 / Float64(t_1 + sqrt(x)))))) + Float64(Float64(1.0 / t_2) + Float64(t_4 - sqrt(t))));
	else
		tmp = Float64(Float64(t_1 + Float64(t_3 - sqrt(x))) + Float64(Float64(1.0 / Float64(t_4 + sqrt(t))) - Float64(-1.0 / t_2)));
	end
	return 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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(Exp[N[Log[1 + N[(t$95$3 + N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + N[(N[(1.0 / t$95$2), $MachinePrecision] + N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.1
Target	0.4
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 59.9

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

   2. Simplified 59.9

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

      [Start] 59.9	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 59.9	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 59.9	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 60.0	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 60.0	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 59.9	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 59.9	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 57.7

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

   4. Simplified 57.7

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

      [Start] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(1 + \left(z - z\right)\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\left(z - z\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{1} \cdot \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   5. Applied egg-rr 57.7

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

   6. Simplified 57.7

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

      [Start] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 57.7	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Applied egg-rr 56.5

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

   8. Applied egg-rr 53.3

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

   9. Simplified 0.4

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

      [Start] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-commutative [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \left(x + \left(1 - x\right)\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*l/ [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1 \cdot \left(x + \left(1 - x\right)\right)}{\sqrt{x + 1} + \sqrt{x}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-*r/ [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{1 \cdot \frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{-1}{-1}} \cdot \frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{-1}}{-1} \cdot \frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{-1}{\color{blue}{-1}} \cdot \frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      times-frac [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(-1\right) \cdot \left(x + \left(1 - x\right)\right)}{\left(-1\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{-1} \cdot \left(x + \left(1 - x\right)\right)}{\left(-1\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      neg-mul-1 [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{-\left(x + \left(1 - x\right)\right)}}{\left(-1\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-/r* [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\frac{-\left(x + \left(1 - x\right)\right)}{-1}}{\sqrt{x + 1} + \sqrt{x}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      neg-mul-1 [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\frac{\color{blue}{-1 \cdot \left(x + \left(1 - x\right)\right)}}{-1}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [<=] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\frac{\color{blue}{\left(-1\right)} \cdot \left(x + \left(1 - x\right)\right)}{-1}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-commutative [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\frac{\color{blue}{\left(x + \left(1 - x\right)\right) \cdot \left(-1\right)}}{-1}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-/l* [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{\frac{x + \left(1 - x\right)}{\frac{-1}{-1}}}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\frac{x + \left(1 - x\right)}{\frac{\color{blue}{-1}}{-1}}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\frac{x + \left(1 - x\right)}{\frac{-1}{\color{blue}{-1}}}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\frac{x + \left(1 - x\right)}{\color{blue}{1}}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      /-rgt-identity [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 53.3	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 0.4	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 0.4	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 0.4	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.4	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.4	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   if 5.00000000000000024e-5 < (-.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

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

      [Start] 2.1	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
      associate-+l+ [=>] 2.1	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]
      associate-+l- [=>] 2.1	\[ \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate--r- [=>] 2.9	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \sqrt{y}\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [<=] 2.9	\[ \left(\sqrt{x + 1} - \left(\left(\sqrt{x} - \sqrt{y + 1}\right) + \color{blue}{\left(-\left(-\sqrt{y}\right)\right)}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      associate-+l- [=>] 2.1	\[ \left(\sqrt{x + 1} - \color{blue}{\left(\sqrt{x} - \left(\sqrt{y + 1} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{\color{blue}{1 + y}} - \left(-\left(-\sqrt{y}\right)\right)\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      remove-double-neg [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} + \left(-\sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      sub-neg [<=] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{\color{blue}{1 + z}} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 2.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{\color{blue}{1 + t}} - \sqrt{t}\right)\right) \]

   3. Applied egg-rr 1.3

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

   4. Simplified 1.3

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

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

   5. Applied egg-rr 0.5

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

   6. Simplified 0.5

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

      [Start] 0.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(1 + \left(y - y\right)\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-commutative [=>] 0.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\left(\left(y - y\right) + 1\right)} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      +-inverses [=>] 0.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\color{blue}{0} + 1\right) \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      metadata-eval [=>] 0.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{1} \cdot \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
      *-lft-identity [=>] 0.5	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   7. Applied egg-rr 0.3

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

   8. Simplified 0.1

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

      [Start] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\left(1 + t\right) - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) \]
      *-commutative [=>] 0.3	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\left(1 + t\right) - t\right)}\right) \]
      associate--l+ [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(1 + \left(t - t\right)\right)}\right) \]
      distribute-rgt-in [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(t - t\right) \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)}\right) \]
      +-inverses [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{0} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      +-inverses [<=] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}} + \color{blue}{\left(z - z\right)} \cdot \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right) \]
      distribute-rgt-out [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(1 + \left(z - z\right)\right)}\right) \]
      +-commutative [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{\left(\left(z - z\right) + 1\right)}\right) \]
      +-inverses [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\color{blue}{0} + 1\right)\right) \]
      metadata-eval [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}} \cdot \color{blue}{1}\right) \]
      *-rgt-identity [=>] 0.1	\[ \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \frac{-1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	79556

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;\left(\left(t_2 - \sqrt{x}\right) + \left(t_3 - \sqrt{y}\right)\right) + \left(t_1 - \sqrt{z}\right) \leq 0.99999999999998:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \frac{-1}{t_1 + \sqrt{z}}\right) + \left(1 + \frac{1}{t_3 + \sqrt{y}}\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	66244

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.99995:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \frac{-1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \end{array} \]

Alternative 3
Error	1.2
Cost	40004

\[\begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 5
Error	2.6
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 1.05 \cdot 10^{-104}:\\ \;\;\;\;1 + \left(\left(t_2 - \left(\sqrt{z} - t_1\right)\right) + \left(\sqrt{1 + t} - \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;t_3 + \left(\frac{1}{t_1 + \sqrt{z}} + \left(t_2 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_3 + \sqrt{x}}\\ \end{array} \]

Alternative 6
Error	2.5
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 7
Error	5.1
Cost	39620

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\left(1 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - t_1\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	5.3
Cost	33220

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\left(1 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\frac{1}{t_1 + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 9
Error	6.0
Cost	33092

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\left(1 + \left(t_1 + \sqrt{1 + t}\right)\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{z} - t_1\right)\right)\\ \end{array} \]

Alternative 10
Error	9.6
Cost	32904

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} + 2\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 42000000000000:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{y}\right) + \left(\mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 11
Error	9.6
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} + 2\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 70000000000000:\\ \;\;\;\;t_1 + \left(\left(1 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 12
Error	9.5
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} + 2\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 50000000000000:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(t_1 - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 13
Error	9.9
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;z \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} + 2\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 680000000000:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 14
Error	10.7
Cost	20168

\[\begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} + 2\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 1600000000000:\\ \;\;\;\;\left(1 + \left(\sqrt{x + 1} + \sqrt{1 + z}\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15
Error	10.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} + 2\right) - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 7000000000000:\\ \;\;\;\;1 + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 16
Error	10.2
Cost	13508

\[\begin{array}{l} \mathbf{if}\;z \leq 5600000000000:\\ \;\;\;\;1 + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 17
Error	22.1
Cost	13384

\[\begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{elif}\;x \leq 2050000000000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	24.4
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;y \leq 5:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	19.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 20
Error	41.2
Cost	13120

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

Alternative 21
Error	41.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022364 
(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))))