Main:z from

Average Error: 5.2 → 0.6

Time: 24.0s

Precision: binary64

Cost: 65856

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

Math

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

↓

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

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
 (+
  (exp
   (log
    (+
     (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))
     (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))))))
  (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) (- (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) {
	return exp(log(((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (1.0 / (sqrt(x) + sqrt((1.0 + x))))))) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t)));
}

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
    code = exp(log(((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x))))))) + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t)))
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) {
	return Math.exp(Math.log(((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x))))))) + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
}

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):
	return math.exp(math.log(((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x))))))) + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t)))

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)
	return Float64(exp(log(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))))) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))))
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 = code(x, y, z, t)
	tmp = exp(log(((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (1.0 / (sqrt(x) + sqrt((1.0 + x))))))) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t)));
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_] := N[(N[Exp[N[Log[N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	5.2
Target	0.4
Herbie	0.6

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

Derivation

1. Initial program 5.2

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

2. Simplified 5.2

   \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   Proof
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 y 1))) (sqrt.f64 y)))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 12 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 y 1)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 y))))))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 y 1))) (neg.f64 (neg.f64 (sqrt.f64 y)))))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (+.f64 (-.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 y 1))) (Rewrite=> remove-double-neg_binary64 (sqrt.f64 y)))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (Rewrite<= associate--r-_binary64 (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)))) (+.f64 (-.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 12 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (+.f64 (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 z 1))) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 12 points decrease in error
   (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (+.f64 (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (sqrt.f64 (+.f64 z 1)) 0)) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 12 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (+.f64 (-.f64 (+.f64 (sqrt.f64 (+.f64 z 1)) 0) (sqrt.f64 z)) (-.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 t 1))) (sqrt.f64 t)))): 0 points increase in error, 12 points decrease in error
   (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (+.f64 (-.f64 (Rewrite=> +-rgt-identity_binary64 (sqrt.f64 (+.f64 z 1))) (sqrt.f64 z)) (-.f64 (sqrt.f64 (+.f64 t 1)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t 1)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 4.2

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

4. Simplified 4.2

   \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   Proof
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))) (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 5 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))) (+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 0 1)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 5 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))) (+.f64 (*.f64 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 z z)) 1) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (-.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))) (+.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (-.f64 z z))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 3.4

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

6. Simplified 3.4

   \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right)\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   Proof
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 5 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (*.f64 (Rewrite<= metadata-eval (+.f64 0 1)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 5 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (*.f64 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 y y)) 1) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (-.f64 (sqrt.f64 x) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (-.f64 y y))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 3.4

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

8. Applied egg-rr 0.6

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

9. Simplified 0.6

   \[\leadsto e^{\log \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   Proof
   (+.f64 (exp.f64 (log.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x)))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (exp.f64 (log.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x)))))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 5 points decrease in error
   (+.f64 (exp.f64 (log.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (*.f64 (Rewrite<= metadata-eval (+.f64 0 1)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 5 points decrease in error
   (+.f64 (exp.f64 (log.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (*.f64 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 x x)) 1) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (exp.f64 (log.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (-.f64 x x))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))))))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error

10. Final simplification 0.6

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

Alternatives

Alternative 1
Error	1.0
Cost	66116

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

Alternative 2
Error	1.2
Cost	40004

\[\begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+33}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 - \frac{-1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 3
Error	1.7
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 1.32 \cdot 10^{-55}:\\ \;\;\;\;\left(\frac{1}{t_2 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+33}:\\ \;\;\;\;t_1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_2 - \left(\sqrt{x} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \]

Alternative 4
Error	1.9
Cost	39876

\[\begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 5
Error	2.3
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 1.32 \cdot 10^{-55}:\\ \;\;\;\;\left(\frac{1}{t_1 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + 2\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 6
Error	6.1
Cost	26568

\[\begin{array}{l} \mathbf{if}\;z \leq 10^{-49}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 1450000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\\ \end{array} \]

Alternative 7
Error	6.2
Cost	26564

\[\begin{array}{l} \mathbf{if}\;t \leq 38000000000000:\\ \;\;\;\;2 + \left(\left(1 + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 8
Error	6.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 2400000000000:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 9
Error	6.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 3100000000000:\\ \;\;\;\;\left(\sqrt{1 + z} + 2\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 10
Error	24.5
Cost	13380

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 1.6:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	8.9
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 0.4:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 12
Error	22.8
Cost	13248

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

Alternative 13
Error	41.1
Cost	13120

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

Alternative 14
Error	41.5
Cost	6848

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

Alternative 15
Error	41.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(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))))