Average Error: 5.2 → 0.1
Time: 21.7s
Precision: binary64
Cost: 66244
\[ \begin{array}{c}[x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \end{array} \]
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := \frac{1}{t_2 + \sqrt{y}}\\ t_4 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ \mathbf{if}\;t_2 - \sqrt{y} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1} + \left(t_3 + t_4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(t_3 + \left(t_4 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]
(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (/ 1.0 (+ t_2 (sqrt y))))
        (t_4 (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))))
   (if (<= (- t_2 (sqrt y)) 5e-8)
     (+ (/ 1.0 (+ (sqrt x) t_1)) (+ t_3 t_4))
     (+
      (- t_1 (sqrt x))
      (+ t_3 (+ t_4 (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t)))))))))
double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double t_3 = 1.0 / (t_2 + sqrt(y));
	double t_4 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	double tmp;
	if ((t_2 - sqrt(y)) <= 5e-8) {
		tmp = (1.0 / (sqrt(x) + t_1)) + (t_3 + t_4);
	} else {
		tmp = (t_1 - sqrt(x)) + (t_3 + (t_4 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))));
	}
	return tmp;
}

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 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = 1.0d0 / (t_2 + sqrt(y))
    t_4 = 1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))
    if ((t_2 - sqrt(y)) <= 5d-8) then
        tmp = (1.0d0 / (sqrt(x) + t_1)) + (t_3 + t_4)
    else
        tmp = (t_1 - sqrt(x)) + (t_3 + (t_4 + (1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t)))))
    end if
    code = tmp
end function

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}
↓
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = 1.0 / (t_2 + Math.sqrt(y));
	double t_4 = 1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z));
	double tmp;
	if ((t_2 - Math.sqrt(y)) <= 5e-8) {
		tmp = (1.0 / (Math.sqrt(x) + t_1)) + (t_3 + t_4);
	} else {
		tmp = (t_1 - Math.sqrt(x)) + (t_3 + (t_4 + (1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t)))));
	}
	return tmp;
}

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
↓
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y))
	t_3 = 1.0 / (t_2 + math.sqrt(y))
	t_4 = 1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))
	tmp = 0
	if (t_2 - math.sqrt(y)) <= 5e-8:
		tmp = (1.0 / (math.sqrt(x) + t_1)) + (t_3 + t_4)
	else:
		tmp = (t_1 - math.sqrt(x)) + (t_3 + (t_4 + (1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t)))))
	return tmp

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end
↓
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(1.0 / Float64(t_2 + sqrt(y)))
	t_4 = Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))
	tmp = 0.0
	if (Float64(t_2 - sqrt(y)) <= 5e-8)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(t_3 + t_4));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(t_3 + Float64(t_4 + Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))))));
	end
	return tmp
end

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end
↓
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y));
	t_3 = 1.0 / (t_2 + sqrt(y));
	t_4 = 1.0 / (sqrt((1.0 + z)) + sqrt(z));
	tmp = 0.0;
	if ((t_2 - sqrt(y)) <= 5e-8)
		tmp = (1.0 / (sqrt(x) + t_1)) + (t_3 + t_4);
	else
		tmp = (t_1 - sqrt(x)) + (t_3 + (t_4 + (1.0 / (sqrt((1.0 + t)) + sqrt(t)))));
	end
	tmp_2 = tmp;
end

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$4 + N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)
↓
\begin{array}{l}
t_1 := \sqrt{1 + x}\\
t_2 := \sqrt{1 + y}\\
t_3 := \frac{1}{t_2 + \sqrt{y}}\\
t_4 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\
\mathbf{if}\;t_2 - \sqrt{y} \leq 5 \cdot 10^{-8}:\\
\;\;\;\;\frac{1}{\sqrt{x} + t_1} + \left(t_3 + t_4\right)\\

\mathbf{else}:\\
\;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(t_3 + \left(t_4 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.2
Target0.4
Herbie0.1
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 4.9999999999999998e-8

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Applied egg-rr3.6

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

    4. Simplified3.6

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

    5. Applied egg-rr3.3

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

    6. Simplified0.4

      \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      Proof
      (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (-.f64 1 0)) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 y y))) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 y) y)) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 63 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 y (-.f64 1 y))) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 y (-.f64 1 y)) 1)) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 y (-.f64 1 y)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))))): 0 points increase in error, 0 points decrease in error

    7. Applied egg-rr0.1

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

    8. Simplified0.1

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

    9. Taylor expanded in t around inf 0.1

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

    if 4.9999999999999998e-8 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

    1. Initial program 2.0

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

    2. Simplified2.0

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

    3. Applied egg-rr0.9

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

    4. Simplified0.9

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

    5. Applied egg-rr0.5

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

    6. Simplified0.5

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      Proof
      (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (-.f64 1 0)) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 y y))) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 y) y)) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 63 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 y (-.f64 1 y))) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 y (-.f64 1 y)) 1)) (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 y (-.f64 1 y)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))))): 0 points increase in error, 0 points decrease in error

    7. Applied egg-rr0.1

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

    8. Simplified0.1

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\frac{1}{\sqrt{1 + t} + \sqrt{t}}}\right)\right) \]
      Proof
      (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (+.f64 1 0)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 t t))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 t)) (sqrt.f64 t)))): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.1
Cost66116
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 - \sqrt{y}\\ t_4 := \frac{1}{\sqrt{1 + z} + \sqrt{z}}\\ \mathbf{if}\;t_3 \leq 0.996:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1} + \left(\frac{1}{t_2 + \sqrt{y}} + t_4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\left(t_4 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right) + t_3\right)\\ \end{array} \]
Alternative 2
Error0.2
Cost53060
\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t_1 - \sqrt{z}\\ t_3 := \sqrt{1 + y}\\ \mathbf{if}\;t_2 \leq 0.02:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{t_3 + \sqrt{y}} + \frac{1}{t_1 + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(t_3 - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + t_2\right)\right)\\ \end{array} \]
Alternative 3
Error0.4
Cost53056
\[\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
Alternative 4
Error0.6
Cost39876
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 1.42 \cdot 10^{+32}:\\ \;\;\;\;1 + \left(\left(\frac{1}{t_2 + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(t_1 - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{t_1 + \sqrt{y}} + \left(t_2 - \sqrt{z}\right)\right)\\ \end{array} \]
Alternative 5
Error5.9
Cost39744
\[1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) \]
Alternative 6
Error5.3
Cost39744
\[1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \]
Alternative 7
Error6.1
Cost39616
\[1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) \]
Alternative 8
Error6.5
Cost26696
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]
Alternative 9
Error6.6
Cost26696
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 86000000000000:\\ \;\;\;\;t_1 + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]
Alternative 10
Error6.9
Cost13512
\[\begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-44}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + 3\\ \mathbf{elif}\;z \leq 255000000000:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
Alternative 11
Error10.2
Cost13380
\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
Alternative 12
Error22.8
Cost13248
\[1 + \left(\sqrt{1 + y} - \sqrt{y}\right) \]
Alternative 13
Error54.6
Cost6848
\[\left(1 + y \cdot 0.5\right) - \sqrt{y} \]
Alternative 14
Error55.7
Cost6592
\[1 - \sqrt{y} \]

Error

Reproduce

herbie shell --seed 2022330 
(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))))