Average Error: 5.4 → 3.9
Time: 42.2s
Precision: binary64
Cost: 105288
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t_1 + \sqrt{y}\\ t_3 := \sqrt{x + 1}\\ t_4 := t_3 + \sqrt{x}\\ \left(\left(\begin{array}{l} \mathbf{if}\;t_4 \ne 0:\\ \;\;\;\;\frac{1}{t_4}\\ \mathbf{else}:\\ \;\;\;\;t_3 - \sqrt{x}\\ \end{array} + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{{t_1}^{2} - {\left(\sqrt{y}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{y}\\ \end{array}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]
(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 (+ y 1.0)))
        (t_2 (+ t_1 (sqrt y)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ t_3 (sqrt x))))
   (+
    (+
     (+
      (if (!= t_4 0.0) (/ 1.0 t_4) (- t_3 (sqrt x)))
      (if (!= t_2 0.0)
        (/ (- (pow t_1 2.0) (pow (sqrt y) 2.0)) t_2)
        (- t_1 (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 (((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((y + 1.0));
	double t_2 = t_1 + sqrt(y);
	double t_3 = sqrt((x + 1.0));
	double t_4 = t_3 + sqrt(x);
	double tmp;
	if (t_4 != 0.0) {
		tmp = 1.0 / t_4;
	} else {
		tmp = t_3 - sqrt(x);
	}
	double tmp_1;
	if (t_2 != 0.0) {
		tmp_1 = (pow(t_1, 2.0) - pow(sqrt(y), 2.0)) / t_2;
	} else {
		tmp_1 = t_1 - sqrt(y);
	}
	return ((tmp + tmp_1) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

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
    real(8) :: tmp_1
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 + sqrt(y)
    t_3 = sqrt((x + 1.0d0))
    t_4 = t_3 + sqrt(x)
    if (t_4 /= 0.0d0) then
        tmp = 1.0d0 / t_4
    else
        tmp = t_3 - sqrt(x)
    end if
    if (t_2 /= 0.0d0) then
        tmp_1 = ((t_1 ** 2.0d0) - (sqrt(y) ** 2.0d0)) / t_2
    else
        tmp_1 = t_1 - sqrt(y)
    end if
    code = ((tmp + tmp_1) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
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((y + 1.0));
	double t_2 = t_1 + Math.sqrt(y);
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = t_3 + Math.sqrt(x);
	double tmp;
	if (t_4 != 0.0) {
		tmp = 1.0 / t_4;
	} else {
		tmp = t_3 - Math.sqrt(x);
	}
	double tmp_1;
	if (t_2 != 0.0) {
		tmp_1 = (Math.pow(t_1, 2.0) - Math.pow(Math.sqrt(y), 2.0)) / t_2;
	} else {
		tmp_1 = t_1 - Math.sqrt(y);
	}
	return ((tmp + tmp_1) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

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((y + 1.0))
	t_2 = t_1 + math.sqrt(y)
	t_3 = math.sqrt((x + 1.0))
	t_4 = t_3 + math.sqrt(x)
	tmp = 0
	if t_4 != 0.0:
		tmp = 1.0 / t_4
	else:
		tmp = t_3 - math.sqrt(x)
	tmp_1 = 0
	if t_2 != 0.0:
		tmp_1 = (math.pow(t_1, 2.0) - math.pow(math.sqrt(y), 2.0)) / t_2
	else:
		tmp_1 = t_1 - math.sqrt(y)
	return ((tmp + tmp_1) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

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(y + 1.0))
	t_2 = Float64(t_1 + sqrt(y))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(t_3 + sqrt(x))
	tmp = 0.0
	if (t_4 != 0.0)
		tmp = Float64(1.0 / t_4);
	else
		tmp = Float64(t_3 - sqrt(x));
	end
	tmp_1 = 0.0
	if (t_2 != 0.0)
		tmp_1 = Float64(Float64((t_1 ^ 2.0) - (sqrt(y) ^ 2.0)) / t_2);
	else
		tmp_1 = Float64(t_1 - sqrt(y));
	end
	return Float64(Float64(Float64(tmp + tmp_1) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
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_3 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = t_1 + sqrt(y);
	t_3 = sqrt((x + 1.0));
	t_4 = t_3 + sqrt(x);
	tmp = 0.0;
	if (t_4 ~= 0.0)
		tmp = 1.0 / t_4;
	else
		tmp = t_3 - sqrt(x);
	end
	tmp_2 = 0.0;
	if (t_2 ~= 0.0)
		tmp_2 = ((t_1 ^ 2.0) - (sqrt(y) ^ 2.0)) / t_2;
	else
		tmp_2 = t_1 - sqrt(y);
	end
	tmp_3 = ((tmp + tmp_2) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(If[Unequal[t$95$4, 0.0], N[(1.0 / t$95$4), $MachinePrecision], N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]] + If[Unequal[t$95$2, 0.0], N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Power[N[Sqrt[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$1 - 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]]]]]

\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{y + 1}\\
t_2 := t_1 + \sqrt{y}\\
t_3 := \sqrt{x + 1}\\
t_4 := t_3 + \sqrt{x}\\
\left(\left(\begin{array}{l}
\mathbf{if}\;t_4 \ne 0:\\
\;\;\;\;\frac{1}{t_4}\\

\mathbf{else}:\\
\;\;\;\;t_3 - \sqrt{x}\\


\end{array} + \begin{array}{l}
\mathbf{if}\;t_2 \ne 0:\\
\;\;\;\;\frac{{t_1}^{2} - {\left(\sqrt{y}\right)}^{2}}{t_2}\\

\mathbf{else}:\\
\;\;\;\;t_1 - \sqrt{y}\\


\end{array}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)
\end{array}

Error

Target

Original5.4
Target1.5
Herbie3.9
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

Derivation

  1. Initial program 5.4

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

  2. Applied egg-rr5.3

    \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sqrt{y + 1} + \sqrt{y} \ne 0:\\ \;\;\;\;\frac{{\left(\sqrt{y + 1}\right)}^{2} - {\left(\sqrt{y}\right)}^{2}}{\sqrt{y + 1} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + 1} - \sqrt{y}\\ } \end{array}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

  3. Applied egg-rr5.2

    \[\leadsto \left(\left(\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sqrt{x + 1} + \sqrt{x} \ne 0:\\ \;\;\;\;\frac{{\left(\sqrt{x + 1}\right)}^{2} - {\left(\sqrt{x}\right)}^{2}}{\sqrt{x + 1} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ } \end{array}} + \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} + \sqrt{y} \ne 0:\\ \;\;\;\;\frac{{\left(\sqrt{y + 1}\right)}^{2} - {\left(\sqrt{y}\right)}^{2}}{\sqrt{y + 1} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + 1} - \sqrt{y}\\ \end{array}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

  4. Taylor expanded in x around 0 3.9

    \[\leadsto \left(\left(\begin{array}{l} \mathbf{if}\;\sqrt{x + 1} + \sqrt{x} \ne 0:\\ \;\;\;\;\frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} + \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} + \sqrt{y} \ne 0:\\ \;\;\;\;\frac{{\left(\sqrt{y + 1}\right)}^{2} - {\left(\sqrt{y}\right)}^{2}}{\sqrt{y + 1} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + 1} - \sqrt{y}\\ \end{array}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

Alternatives

Alternative 1
Error5.3
Cost104900
\[\begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t_1 + \sqrt{y}\\ \left(\left({\left({\left(\sqrt{x + 1} - \sqrt{x}\right)}^{3}\right)}^{0.3333333333333333} + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{{t_1}^{2} - {\left(\sqrt{y}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{y}\\ \end{array}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]
Alternative 2
Error5.3
Cost91972
\[\begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := t_1 + \sqrt{t}\\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{{t_1}^{2} - {\left(\sqrt{t}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{t}\\ \end{array} \end{array} \]
Alternative 3
Error5.3
Cost91972
\[\begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t_1 + \sqrt{z}\\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{{t_1}^{2} - {\left(\sqrt{z}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{z}\\ \end{array}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]
Alternative 4
Error5.3
Cost91972
\[\begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t_1 + \sqrt{y}\\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{{t_1}^{2} - {\left(\sqrt{y}\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{y}\\ \end{array}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]
Alternative 5
Error5.4
Cost52672
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
Alternative 6
Error29.3
Cost39884
\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 6 \cdot 10^{-160}:\\ \;\;\;\;t_1 + 2\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;1 + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\sqrt{z + 1} + \left(\sqrt{t + 1} - \left(\sqrt{t} + \sqrt{z}\right)\right)\right)\\ \end{array} \]
Alternative 7
Error21.0
Cost39884
\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;t_1 + 2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;1 + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right)\\ \end{array} \]
Alternative 8
Error15.6
Cost39880
\[\begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t_1 - \sqrt{z}\\ t_3 := \sqrt{t + 1}\\ t_4 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(1 + t_4\right) + t_2\right) + \left(t_3 - \sqrt{t}\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+53}:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t_4\right) + t_2\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 + \left(t_3 - \left(\sqrt{t} + \sqrt{z}\right)\right)\right)\\ \end{array} \]
Alternative 9
Error16.1
Cost39748
\[\begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{t + 1}\\ \mathbf{if}\;x \leq 650000000:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(t_1 - \sqrt{z}\right)\right) + \left(t_2 - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 + \left(t_2 - \left(\sqrt{t} + \sqrt{z}\right)\right)\right)\\ \end{array} \]
Alternative 10
Error34.1
Cost28016
\[\begin{array}{l} t_1 := \left(1 + \sqrt{1 + t}\right) - \sqrt{t}\\ t_2 := \sqrt{1 + z}\\ t_3 := \left(1 + t_2\right) - \sqrt{z}\\ t_4 := 1 + t_3\\ t_5 := \sqrt{1 + x} - \sqrt{x}\\ t_6 := t_5 + t_3\\ \mathbf{if}\;t \leq 7 \cdot 10^{-256}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-69}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+26}:\\ \;\;\;\;t_5 + t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+58}:\\ \;\;\;\;t_5 + \left(t_2 - \sqrt{z}\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+122}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+175}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+187}:\\ \;\;\;\;1 + \left(\sqrt{z + 1} + \left(\sqrt{t + 1} - \left(\sqrt{t} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+217}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+251}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+253}:\\ \;\;\;\;\left(t_2 + 1\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]
Alternative 11
Error36.2
Cost27096
\[\begin{array}{l} t_1 := \left(1 + \sqrt{1 + t}\right) - \sqrt{t}\\ t_2 := \sqrt{1 + z}\\ t_3 := 1 + \left(\left(1 + t_2\right) - \sqrt{z}\right)\\ \mathbf{if}\;t \leq 3.4 \cdot 10^{-256}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+58}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t_2 - \sqrt{z}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+122}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+175}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+221}:\\ \;\;\;\;\left(t_2 + 1\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 12
Error34.8
Cost26964
\[\begin{array}{l} t_1 := \left(1 + \sqrt{1 + t}\right) - \sqrt{t}\\ t_2 := \sqrt{1 + z}\\ t_3 := 1 + \left(\left(1 + t_2\right) - \sqrt{z}\right)\\ t_4 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t \leq 7.5 \cdot 10^{-256}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+27}:\\ \;\;\;\;t_4 + t_1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+58}:\\ \;\;\;\;t_4 + \left(t_2 - \sqrt{z}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+122}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+175}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+221}:\\ \;\;\;\;\left(t_2 + 1\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 13
Error34.3
Cost26696
\[\begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-199}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + 2\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-35}:\\ \;\;\;\;1 + \left(\sqrt{z + 1} + \left(\sqrt{t + 1} - \left(\sqrt{t} + \sqrt{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+204}:\\ \;\;\;\;1 + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \end{array} \]
Alternative 14
Error36.7
Cost14696
\[\begin{array}{l} t_1 := \left(1 + \sqrt{1 + t}\right) - \sqrt{t}\\ t_2 := \sqrt{1 + z}\\ t_3 := \left(t_2 + 1\right) - \sqrt{z}\\ t_4 := 1 + \left(\left(1 + t_2\right) - \sqrt{z}\right)\\ \mathbf{if}\;t \leq 2.3 \cdot 10^{-256}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.56 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{+31}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+122}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+175}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+221}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]
Alternative 15
Error35.3
Cost13908
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \left(t_1 - \sqrt{x}\right) + 2\\ t_3 := \left(1 + t_1\right) - \sqrt{x}\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{-115}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 16
Error34.3
Cost13776
\[\begin{array}{l} t_1 := \left(\sqrt{1 + x} - \sqrt{x}\right) + 2\\ \mathbf{if}\;t \leq 3200000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+112}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.55 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+278}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 17
Error35.3
Cost13644
\[\begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + 2\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+26}:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+204}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\\ \end{array} \]
Alternative 18
Error34.4
Cost13512
\[\begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + 2\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+207}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{1 + t}\right) - \sqrt{t}\\ \end{array} \]
Alternative 19
Error41.6
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2023010 
(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))))