Average Error: 5.6 → 0.1
Time: 25.1s
Precision: binary64
Cost: 65988
\[ \begin{array}{c}[x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \end{array} \]
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := t_2 - \sqrt{z}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;t_3 \leq 0.02:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{z}} + \left(\frac{1}{t_1 + \sqrt{x}} + \frac{1}{t_4 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\left(t_4 - \sqrt{y}\right) + \left(t_3 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]
(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 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= t_3 0.02)
     (+
      (/ 1.0 (+ t_2 (sqrt z)))
      (+ (/ 1.0 (+ t_1 (sqrt x))) (/ 1.0 (+ t_4 (sqrt y)))))
     (+
      (- t_1 (sqrt x))
      (+ (- t_4 (sqrt y)) (+ t_3 (/ 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 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 0.02) {
		tmp = (1.0 / (t_2 + sqrt(z))) + ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_4 + sqrt(y))));
	} else {
		tmp = (t_1 - sqrt(x)) + ((t_4 - sqrt(y)) + (t_3 + (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 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + y))
    if (t_3 <= 0.02d0) then
        tmp = (1.0d0 / (t_2 + sqrt(z))) + ((1.0d0 / (t_1 + sqrt(x))) + (1.0d0 / (t_4 + sqrt(y))))
    else
        tmp = (t_1 - sqrt(x)) + ((t_4 - sqrt(y)) + (t_3 + (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 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 0.02) {
		tmp = (1.0 / (t_2 + Math.sqrt(z))) + ((1.0 / (t_1 + Math.sqrt(x))) + (1.0 / (t_4 + Math.sqrt(y))));
	} else {
		tmp = (t_1 - Math.sqrt(x)) + ((t_4 - Math.sqrt(y)) + (t_3 + (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 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if t_3 <= 0.02:
		tmp = (1.0 / (t_2 + math.sqrt(z))) + ((1.0 / (t_1 + math.sqrt(x))) + (1.0 / (t_4 + math.sqrt(y))))
	else:
		tmp = (t_1 - math.sqrt(x)) + ((t_4 - math.sqrt(y)) + (t_3 + (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 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_3 <= 0.02)
		tmp = Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(Float64(1.0 / Float64(t_1 + sqrt(x))) + Float64(1.0 / Float64(t_4 + sqrt(y)))));
	else
		tmp = Float64(Float64(t_1 - sqrt(x)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_3 + 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 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_3 <= 0.02)
		tmp = (1.0 / (t_2 + sqrt(z))) + ((1.0 / (t_1 + sqrt(x))) + (1.0 / (t_4 + sqrt(y))));
	else
		tmp = (t_1 - sqrt(x)) + ((t_4 - sqrt(y)) + (t_3 + (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 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.02], N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + 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 + z}\\
t_3 := t_2 - \sqrt{z}\\
t_4 := \sqrt{1 + y}\\
\mathbf{if}\;t_3 \leq 0.02:\\
\;\;\;\;\frac{1}{t_2 + \sqrt{z}} + \left(\frac{1}{t_1 + \sqrt{x}} + \frac{1}{t_4 + \sqrt{y}}\right)\\

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

Original5.6
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 z 1)) (sqrt.f64 z)) < 0.0200000000000000004

    1. Initial program 7.7

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

    2. Simplified7.7

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)} \]
      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)))): 1 points increase in error, 0 points decrease in error

    3. Applied egg-rr3.3

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

    4. Simplified3.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      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-rr2.7

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

    6. Simplified1.4

      \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      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))): 74 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-rr0.2

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

    8. Simplified0.2

      \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) \]
      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 0.2

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

    10. Simplified0.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} \]
      Proof
      (+.f64 (/.f64 1 (+.f64 (sqrt.f64 z) (sqrt.f64 (+.f64 1 z)))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (/.f64 1 (+.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z)))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (/.f64 1 (+.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 z)) (sqrt.f64 z))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x)))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))))): 7 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (sqrt.f64 z) (sqrt.f64 (+.f64 1 z))))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x)))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 z) (sqrt.f64 (+.f64 1 z)))) (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 1 x)))))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 1 x)))) (/.f64 1 (+.f64 (sqrt.f64 z) (sqrt.f64 (+.f64 1 z)))))) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 y)) (sqrt.f64 y))) (+.f64 (/.f64 1 (+.f64 (sqrt.f64 x) (sqrt.f64 (+.f64 1 x)))) (/.f64 1 (+.f64 (sqrt.f64 z) (sqrt.f64 (+.f64 1 z))))))): 0 points increase in error, 0 points decrease in error

    if 0.0200000000000000004 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

    1. Initial program 0.8

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

    2. Simplified0.8

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)} \]
      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)))): 1 points increase in error, 0 points decrease in error

    3. Applied egg-rr0.1

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

    4. Simplified0.1

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0.02:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.4
Cost53056
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right) \]
Alternative 2
Error0.5
Cost40004
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 6.8 \cdot 10^{+27}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(t_1 - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\frac{1}{t_2 + \sqrt{x}} + \frac{1}{t_1 + \sqrt{y}}\right)\\ \end{array} \]
Alternative 3
Error6.6
Cost39880
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.75 \cdot 10^{-114}:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-97}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) + \left(t_1 - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \left(1 + \mathsf{hypot}\left(1, \sqrt{z}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]
Alternative 4
Error1.4
Cost39876
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 6.8 \cdot 10^{+27}:\\ \;\;\;\;\left(t_2 - \sqrt{x}\right) + \left(\left(t_1 - \sqrt{y}\right) + \left(1 + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}} + \left(\frac{1}{t_1 + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]
Alternative 5
Error2.8
Cost39748
\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;z \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;1 + \left(\left(t_1 + t_2\right) + \left(\left(\sqrt{1 + t} - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(t_2 + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]
Alternative 6
Error2.9
Cost39748
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
Alternative 7
Error6.0
Cost39620
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(t_1 + t_2\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}} + \left(t_2 - \sqrt{y}\right)\\ \end{array} \]
Alternative 8
Error6.2
Cost26568
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.4 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\left(t_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]
Alternative 9
Error6.1
Cost26568
\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;\left(\sqrt{1 + y} + \left(t_1 - \sqrt{y}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \]
Alternative 10
Error6.1
Cost26564
\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
Alternative 11
Error6.3
Cost19908
\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+24}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
Alternative 12
Error6.6
Cost13512
\[\begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+24}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
Alternative 13
Error12.5
Cost13380
\[\begin{array}{l} \mathbf{if}\;z \leq 0.45:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
Alternative 14
Error10.2
Cost13380
\[\begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
Alternative 15
Error30.0
Cost13252
\[\begin{array}{l} \mathbf{if}\;y \leq 4:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
Alternative 16
Error30.4
Cost6980
\[\begin{array}{l} \mathbf{if}\;y \leq 5.1:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]
Alternative 17
Error30.8
Cost6724
\[\begin{array}{l} \mathbf{if}\;y \leq 4:\\ \;\;\;\;3 - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 18
Error41.8
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022329 
(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))))