Main:z from

Percentage Accurate: 91.6% → 99.3%

Time: 30.4s

Alternatives: 17

Speedup: 1.3×

Specification

Math

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

FPCore

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

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));
}

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

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));
}

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))

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

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

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]

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

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));
}

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

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));
}

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))

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

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

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]

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}\\ \mathsf{fma}\left(t_1, t_1, \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) -0.5)))
   (+
    (fma
     t_1
     t_1
     (+
      (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
      (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y)))))
    (- (sqrt (+ 1.0 t)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = pow((sqrt((1.0 + x)) + sqrt(x)), -0.5);
	return fma(t_1, t_1, ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (1.0 / (sqrt((1.0 + y)) + sqrt(y))))) + (sqrt((1.0 + t)) - sqrt(t));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) + sqrt(x)) ^ -0.5
	return Float64(fma(t_1, t_1, Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Power[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, N[(N[(t$95$1 * t$95$1 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 91.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.2%

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

   2. +-commutative 91.2%

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

   3. +-commutative 91.2%

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

   4. +-commutative 91.2%

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

   5. +-commutative 91.2%

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

   6. +-commutative 91.2%

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

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
4. Step-by-step derivation

   1. flip-- 91.2%

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

   2. add-sqr-sqrt 73.2%

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

   3. +-commutative 73.2%

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

   4. add-sqr-sqrt 91.4%

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

   5. +-commutative 91.4%

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

5. Applied egg-rr 91.4%

   \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. associate--l+ 92.8%

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

   2. +-inverses 92.8%

      \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

   3. metadata-eval 92.8%

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

7. Simplified 92.8%

   \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
8. Step-by-step derivation

   1. flip-- 92.8%

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

   2. add-sqr-sqrt 72.2%

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

   3. add-sqr-sqrt 93.1%

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

9. Applied egg-rr 93.1%

   \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
10. Step-by-step derivation

    1. associate--l+ 94.6%

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

    2. +-inverses 94.6%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    3. metadata-eval 94.6%

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

11. Simplified 94.6%

    \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
12. Step-by-step derivation

    1. flip-- 94.8%

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

    2. add-sqr-sqrt 68.7%

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

    3. add-sqr-sqrt 94.9%

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

13. Applied egg-rr 94.9%

    \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
14. Step-by-step derivation

    1. associate--l+ 96.8%

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

    2. +-inverses 96.8%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    3. metadata-eval 96.8%

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

15. Simplified 96.8%

    \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
16. Step-by-step derivation

    1. inv-pow 96.8%

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

    2. +-commutative 96.8%

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

    3. metadata-eval 96.8%

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

    4. sqrt-pow2 96.8%

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

    5. sqr-pow 96.8%

       \[\leadsto \left(\color{blue}{{\left(\sqrt{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(\sqrt{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(\frac{-2}{2}\right)}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    6. fma-def 96.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(\frac{-2}{2}\right)}, {\left(\sqrt{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(\frac{-2}{2}\right)}, \frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + z} + \sqrt{z}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

17. Applied egg-rr 96.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}, {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}, \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

18. Final simplification 96.8%

    \[\leadsto \mathsf{fma}\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}, {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}, \frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := t_1 - \sqrt{y}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_2 \leq 0.1:\\ \;\;\;\;t_4 + \left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{t_3 + \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 + \left(\left(t_3 - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_2 0.1)
     (+ t_4 (+ (/ 1.0 (+ t_1 (sqrt y))) (/ 1.0 (+ t_3 (sqrt x)))))
     (+
      t_4
      (+ (- t_3 (sqrt x)) (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = t_1 - sqrt(y);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_2 <= 0.1) {
		tmp = t_4 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (t_3 + sqrt(x))));
	} else {
		tmp = t_4 + ((t_3 - sqrt(x)) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_2));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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 + y))
    t_2 = t_1 - sqrt(y)
    t_3 = sqrt((1.0d0 + x))
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_2 <= 0.1d0) then
        tmp = t_4 + ((1.0d0 / (t_1 + sqrt(y))) + (1.0d0 / (t_3 + sqrt(x))))
    else
        tmp = t_4 + ((t_3 - sqrt(x)) + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + t_2))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = Math.sqrt((1.0 + x));
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 0.1) {
		tmp = t_4 + ((1.0 / (t_1 + Math.sqrt(y))) + (1.0 / (t_3 + Math.sqrt(x))));
	} else {
		tmp = t_4 + ((t_3 - Math.sqrt(x)) + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + t_2));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = t_1 - math.sqrt(y)
	t_3 = math.sqrt((1.0 + x))
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 0.1:
		tmp = t_4 + ((1.0 / (t_1 + math.sqrt(y))) + (1.0 / (t_3 + math.sqrt(x))))
	else:
		tmp = t_4 + ((t_3 - math.sqrt(x)) + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 0.1)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + Float64(1.0 / Float64(t_3 + sqrt(x)))));
	else
		tmp = Float64(t_4 + Float64(Float64(t_3 - sqrt(x)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + t_2)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = t_1 - sqrt(y);
	t_3 = sqrt((1.0 + x));
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 0.1)
		tmp = t_4 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (t_3 + sqrt(x))));
	else
		tmp = t_4 + ((t_3 - sqrt(x)) + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.1], N[(t$95$4 + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)) < 0.10000000000000001

   1. Initial program 85.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 85.7%

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

      2. +-commutative 85.7%

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

      3. +-commutative 85.7%

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

      4. +-commutative 85.7%

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

      5. +-commutative 85.7%

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

      6. +-commutative 85.7%

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

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 85.7%

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

      2. add-sqr-sqrt 67.5%

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

      3. +-commutative 67.5%

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

      4. add-sqr-sqrt 85.7%

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

      5. +-commutative 85.7%

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

   5. Applied egg-rr 85.7%

      \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 87.8%

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

      2. +-inverses 87.8%

         \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 87.8%

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

   7. Simplified 87.8%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. flip-- 87.9%

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

      2. add-sqr-sqrt 48.4%

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

      3. add-sqr-sqrt 88.5%

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

   9. Applied egg-rr 88.5%

      \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 91.3%

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

       2. +-inverses 91.3%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 91.3%

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

   11. Simplified 91.3%

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

   12. Taylor expanded in z around inf 51.0%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. +-commutative 51.0%

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

   14. Simplified 51.0%

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

   if 0.10000000000000001 < (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.2%

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

      2. +-commutative 97.2%

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

      3. +-commutative 97.2%

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

      4. +-commutative 97.2%

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

      5. +-commutative 97.2%

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

      6. +-commutative 97.2%

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

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 98.4%

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

      2. add-sqr-sqrt 70.7%

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

      3. add-sqr-sqrt 98.6%

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

   5. Applied egg-rr 97.5%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 98.7%

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

      2. +-inverses 98.7%

         \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 98.7%

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

   7. Simplified 97.7%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + y} - \sqrt{y} \leq 0.1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t_2 - \sqrt{x} \leq 0.9999999999999994:\\ \;\;\;\;t_3 + \left(t_1 + \frac{1}{t_2 + \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_2 (sqrt x)) 0.9999999999999994)
     (+ t_3 (+ t_1 (/ 1.0 (+ t_2 (sqrt x)))))
     (+ t_3 (+ 1.0 (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_2 - sqrt(x)) <= 0.9999999999999994) {
		tmp = t_3 + (t_1 + (1.0 / (t_2 + sqrt(x))));
	} else {
		tmp = t_3 + (1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_1));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    t_1 = 1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_2 - sqrt(x)) <= 0.9999999999999994d0) then
        tmp = t_3 + (t_1 + (1.0d0 / (t_2 + sqrt(x))))
    else
        tmp = t_3 + (1.0d0 + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + t_1))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_2 - Math.sqrt(x)) <= 0.9999999999999994) {
		tmp = t_3 + (t_1 + (1.0 / (t_2 + Math.sqrt(x))));
	} else {
		tmp = t_3 + (1.0 + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + t_1));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_2 - math.sqrt(x)) <= 0.9999999999999994:
		tmp = t_3 + (t_1 + (1.0 / (t_2 + math.sqrt(x))))
	else:
		tmp = t_3 + (1.0 + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + t_1))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(x)) <= 0.9999999999999994)
		tmp = Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(t_2 + sqrt(x)))));
	else
		tmp = Float64(t_3 + Float64(1.0 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + t_1)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / (sqrt((1.0 + y)) + sqrt(y));
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_2 - sqrt(x)) <= 0.9999999999999994)
		tmp = t_3 + (t_1 + (1.0 / (t_2 + sqrt(x))));
	else
		tmp = t_3 + (1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.9999999999999994], N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(1.0 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.999999999999999445

   1. Initial program 86.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.5%

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

      2. +-commutative 86.5%

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

      3. +-commutative 86.5%

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

      4. +-commutative 86.5%

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

      5. +-commutative 86.5%

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

      6. +-commutative 86.5%

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

   3. Simplified 86.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 86.5%

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

      2. add-sqr-sqrt 54.7%

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

      3. +-commutative 54.7%

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

      4. add-sqr-sqrt 86.9%

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

      5. +-commutative 86.9%

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

   5. Applied egg-rr 86.9%

      \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 89.4%

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

      2. +-inverses 89.4%

         \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 89.4%

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

   7. Simplified 89.4%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. flip-- 89.4%

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

      2. add-sqr-sqrt 68.9%

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

      3. add-sqr-sqrt 89.6%

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

   9. Applied egg-rr 89.6%

      \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 92.1%

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

       2. +-inverses 92.1%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 92.1%

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

   11. Simplified 92.1%

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

   12. Taylor expanded in z around inf 54.1%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. +-commutative 54.1%

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

   14. Simplified 54.1%

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

   if 0.999999999999999445 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.3%

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

      2. +-commutative 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

      6. +-commutative 97.3%

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

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 97.3%

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

      2. add-sqr-sqrt 97.3%

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

      3. +-commutative 97.3%

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

      4. add-sqr-sqrt 97.3%

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

      5. +-commutative 97.3%

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

   5. Applied egg-rr 97.3%

      \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 97.3%

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

      2. +-inverses 97.3%

         \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 97.3%

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

   7. Simplified 97.3%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. flip-- 97.3%

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

      2. add-sqr-sqrt 76.4%

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

      3. add-sqr-sqrt 97.7%

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

   9. Applied egg-rr 97.7%

      \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 97.9%

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

       2. +-inverses 97.9%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 97.9%

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

   11. Simplified 97.9%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   12. Step-by-step derivation

       1. flip-- 98.4%

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

       2. add-sqr-sqrt 71.2%

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

       3. add-sqr-sqrt 98.4%

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

   13. Applied egg-rr 98.4%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   14. Step-by-step derivation

       1. associate--l+ 98.8%

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

       2. +-inverses 98.8%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 98.8%

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

   15. Simplified 98.8%

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

   16. Taylor expanded in x around 0 98.8%

       \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   17. Step-by-step derivation

       1. +-commutative 98.8%

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

   18. Simplified 98.8%

       \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.9999999999999994:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+
  (- (sqrt (+ 1.0 t)) (sqrt t))
  (+
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z)))
    (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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 = (sqrt((1.0d0 + t)) - sqrt(t)) + (((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) + (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + t)) - math.sqrt(t)) + (((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + t)) - sqrt(t)) + (((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.2%

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

   2. +-commutative 91.2%

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

   3. +-commutative 91.2%

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

   4. +-commutative 91.2%

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

   5. +-commutative 91.2%

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

   6. +-commutative 91.2%

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

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
4. Step-by-step derivation

   1. flip-- 91.2%

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

   2. add-sqr-sqrt 73.2%

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

   3. +-commutative 73.2%

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

   4. add-sqr-sqrt 91.4%

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

   5. +-commutative 91.4%

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

5. Applied egg-rr 91.4%

   \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. associate--l+ 92.8%

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

   2. +-inverses 92.8%

      \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

   3. metadata-eval 92.8%

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

7. Simplified 92.8%

   \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
8. Step-by-step derivation

   1. flip-- 92.8%

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

   2. add-sqr-sqrt 72.2%

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

   3. add-sqr-sqrt 93.1%

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

9. Applied egg-rr 93.1%

   \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
10. Step-by-step derivation

    1. associate--l+ 94.6%

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

    2. +-inverses 94.6%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    3. metadata-eval 94.6%

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

11. Simplified 94.6%

    \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
12. Step-by-step derivation

    1. flip-- 94.8%

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

    2. add-sqr-sqrt 68.7%

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

    3. add-sqr-sqrt 94.9%

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

13. Applied egg-rr 94.9%

    \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
14. Step-by-step derivation

    1. associate--l+ 96.8%

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

    2. +-inverses 96.8%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    3. metadata-eval 96.8%

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

15. Simplified 96.8%

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

16. Final simplification 96.8%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right) + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+
  (- (sqrt (+ 1.0 t)) (sqrt t))
  (+
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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 = (sqrt((1.0d0 + t)) - sqrt(t)) + (((sqrt((1.0d0 + z)) - sqrt(z)) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))) + (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + t)) - math.sqrt(t)) + (((math.sqrt((1.0 + z)) - math.sqrt(z)) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + t)) - sqrt(t)) + (((sqrt((1.0 + z)) - sqrt(z)) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.2%

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

   2. +-commutative 91.2%

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

   3. +-commutative 91.2%

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

   4. +-commutative 91.2%

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

   5. +-commutative 91.2%

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

   6. +-commutative 91.2%

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

3. Simplified 91.2%

   \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
4. Step-by-step derivation

   1. flip-- 91.2%

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

   2. add-sqr-sqrt 73.2%

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

   3. +-commutative 73.2%

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

   4. add-sqr-sqrt 91.4%

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

   5. +-commutative 91.4%

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

5. Applied egg-rr 91.4%

   \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. associate--l+ 92.8%

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

   2. +-inverses 92.8%

      \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

   3. metadata-eval 92.8%

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

7. Simplified 92.8%

   \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
8. Step-by-step derivation

   1. flip-- 92.8%

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

   2. add-sqr-sqrt 72.2%

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

   3. add-sqr-sqrt 93.1%

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

9. Applied egg-rr 93.1%

   \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
10. Step-by-step derivation

    1. associate--l+ 94.6%

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

    2. +-inverses 94.6%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

    3. metadata-eval 94.6%

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

11. Simplified 94.6%

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

12. Final simplification 94.6%

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

Alternative 6: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;\left(1 + \left(t_2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(t_1 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\frac{1}{t_2 + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 5.5e+22)
     (+ (+ 1.0 (+ t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))) (- t_1 (sqrt y)))
     (+
      t_1
      (+ (/ 1.0 (+ t_2 (sqrt y))) (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 5.5e+22) {
		tmp = (1.0 + (t_2 + (sqrt((1.0 + z)) - sqrt(z)))) + (t_1 - sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + y))
    if (z <= 5.5d+22) then
        tmp = (1.0d0 + (t_2 + (sqrt((1.0d0 + z)) - sqrt(z)))) + (t_1 - sqrt(y))
    else
        tmp = t_1 + ((1.0d0 / (t_2 + sqrt(y))) + (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 5.5e+22) {
		tmp = (1.0 + (t_2 + (Math.sqrt((1.0 + z)) - Math.sqrt(z)))) + (t_1 - Math.sqrt(y));
	} else {
		tmp = t_1 + ((1.0 / (t_2 + Math.sqrt(y))) + (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 5.5e+22:
		tmp = (1.0 + (t_2 + (math.sqrt((1.0 + z)) - math.sqrt(z)))) + (t_1 - math.sqrt(y))
	else:
		tmp = t_1 + ((1.0 / (t_2 + math.sqrt(y))) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 5.5e+22)
		tmp = Float64(Float64(1.0 + Float64(t_2 + Float64(sqrt(Float64(1.0 + z)) - sqrt(z)))) + Float64(t_1 - sqrt(y)));
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 5.5e+22)
		tmp = (1.0 + (t_2 + (sqrt((1.0 + z)) - sqrt(z)))) + (t_1 - sqrt(y));
	else
		tmp = t_1 + ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.5e+22], N[(N[(1.0 + N[(t$95$2 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.50000000000000021e22

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 77.4%

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

      4. associate-+l- 64.7%

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

      5. associate-+r- 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in x around 0 28.9%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 28.9%

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

      2. associate--l+ 28.9%

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

   6. Simplified 28.9%

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

   if 5.50000000000000021e22 < z

   1. Initial program 86.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.4%

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

      2. +-commutative 86.4%

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

      3. +-commutative 86.4%

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

      4. +-commutative 86.4%

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

      5. +-commutative 86.4%

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

      6. +-commutative 86.4%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 69.2%

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

      3. +-commutative 69.2%

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

      4. add-sqr-sqrt 86.5%

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

      5. +-commutative 86.5%

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

   5. Applied egg-rr 86.5%

      \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 89.2%

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

      2. +-inverses 89.2%

         \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 89.2%

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

   7. Simplified 89.2%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. flip-- 89.2%

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

      2. add-sqr-sqrt 71.9%

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

      3. add-sqr-sqrt 89.4%

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

   9. Applied egg-rr 89.4%

      \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 92.1%

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

       2. +-inverses 92.1%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 92.1%

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

   11. Simplified 92.1%

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

   12. Taylor expanded in z around inf 92.1%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. +-commutative 92.1%

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

   14. Simplified 92.1%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \end{array} \]

Alternative 7: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;\left(1 + \left(t_2 + \left(\sqrt{1 + z} + \frac{1}{t_1 + \sqrt{t}}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{t}\right) + \left(\frac{1}{t_2 + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 t))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= z 5.5e+22)
     (-
      (+ 1.0 (+ t_2 (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ t_1 (sqrt t))))))
      (+ (sqrt y) (sqrt z)))
     (+
      (- t_1 (sqrt t))
      (+ (/ 1.0 (+ t_2 (sqrt y))) (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (z <= 5.5e+22) {
		tmp = (1.0 + (t_2 + (sqrt((1.0 + z)) + (1.0 / (t_1 + sqrt(t)))))) - (sqrt(y) + sqrt(z));
	} else {
		tmp = (t_1 - sqrt(t)) + ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    t_1 = sqrt((1.0d0 + t))
    t_2 = sqrt((1.0d0 + y))
    if (z <= 5.5d+22) then
        tmp = (1.0d0 + (t_2 + (sqrt((1.0d0 + z)) + (1.0d0 / (t_1 + sqrt(t)))))) - (sqrt(y) + sqrt(z))
    else
        tmp = (t_1 - sqrt(t)) + ((1.0d0 / (t_2 + sqrt(y))) + (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 5.5e+22) {
		tmp = (1.0 + (t_2 + (Math.sqrt((1.0 + z)) + (1.0 / (t_1 + Math.sqrt(t)))))) - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = (t_1 - Math.sqrt(t)) + ((1.0 / (t_2 + Math.sqrt(y))) + (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 5.5e+22:
		tmp = (1.0 + (t_2 + (math.sqrt((1.0 + z)) + (1.0 / (t_1 + math.sqrt(t)))))) - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = (t_1 - math.sqrt(t)) + ((1.0 / (t_2 + math.sqrt(y))) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + t))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 5.5e+22)
		tmp = Float64(Float64(1.0 + Float64(t_2 + Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(t_1 + sqrt(t)))))) - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(Float64(t_1 - sqrt(t)) + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 5.5e+22)
		tmp = (1.0 + (t_2 + (sqrt((1.0 + z)) + (1.0 / (t_1 + sqrt(t)))))) - (sqrt(y) + sqrt(z));
	else
		tmp = (t_1 - sqrt(t)) + ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.5e+22], N[(N[(1.0 + N[(t$95$2 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.50000000000000021e22

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 96.3%

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

      2. associate-+r+ 96.3%

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

      3. associate-+r- 77.4%

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

      4. associate-+l- 64.7%

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

      5. associate-+r- 49.6%

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

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\left(\sqrt{1 + z} - \left(\left(\sqrt{z} - \left(\sqrt{x + 1} - \sqrt{x}\right)\right) - \sqrt{1 + y}\right)\right) - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right)} \]
   4. Step-by-step derivation

      1. flip-- 49.9%

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

      2. add-sqr-sqrt 39.1%

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

      3. add-sqr-sqrt 50.0%

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

   5. Applied egg-rr 50.0%

      \[\leadsto \left(\sqrt{1 + z} - \left(\left(\sqrt{z} - \left(\sqrt{x + 1} - \sqrt{x}\right)\right) - \sqrt{1 + y}\right)\right) - \left(\sqrt{y} - \color{blue}{\frac{\left(1 + t\right) - t}{\sqrt{1 + t} + \sqrt{t}}}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 50.3%

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

      2. +-inverses 50.3%

         \[\leadsto \left(\sqrt{1 + z} - \left(\left(\sqrt{z} - \left(\sqrt{x + 1} - \sqrt{x}\right)\right) - \sqrt{1 + y}\right)\right) - \left(\sqrt{y} - \frac{1 + \color{blue}{0}}{\sqrt{1 + t} + \sqrt{t}}\right) \]

      3. metadata-eval 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in x around 0 29.0%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 29.0%

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

   10. Simplified 29.0%

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

   if 5.50000000000000021e22 < z

   1. Initial program 86.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.4%

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

      2. +-commutative 86.4%

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

      3. +-commutative 86.4%

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

      4. +-commutative 86.4%

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

      5. +-commutative 86.4%

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

      6. +-commutative 86.4%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 86.4%

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

      2. add-sqr-sqrt 69.2%

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

      3. +-commutative 69.2%

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

      4. add-sqr-sqrt 86.5%

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

      5. +-commutative 86.5%

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

   5. Applied egg-rr 86.5%

      \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 89.2%

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

      2. +-inverses 89.2%

         \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 89.2%

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

   7. Simplified 89.2%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. flip-- 89.2%

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

      2. add-sqr-sqrt 71.9%

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

      3. add-sqr-sqrt 89.4%

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

   9. Applied egg-rr 89.4%

      \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 92.1%

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

       2. +-inverses 92.1%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 92.1%

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

   11. Simplified 92.1%

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

   12. Taylor expanded in z around inf 92.1%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. +-commutative 92.1%

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

   14. Simplified 92.1%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{1 + t} + \sqrt{t}}\right)\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \end{array} \]

Alternative 8: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;z \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + t_1\right)\right) + \left(t_2 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{1}{t_1 + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= z 1.5e+30)
     (+ (+ 1.0 (+ (/ 1.0 (+ (sqrt (+ 1.0 z)) (sqrt z))) t_1)) (- t_2 (sqrt y)))
     (+
      t_2
      (+ (/ 1.0 (+ t_1 (sqrt y))) (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (z <= 1.5e+30) {
		tmp = (1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_1)) + (t_2 - sqrt(y));
	} else {
		tmp = t_2 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (z <= 1.5d+30) then
        tmp = (1.0d0 + ((1.0d0 / (sqrt((1.0d0 + z)) + sqrt(z))) + t_1)) + (t_2 - sqrt(y))
    else
        tmp = t_2 + ((1.0d0 / (t_1 + sqrt(y))) + (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (z <= 1.5e+30) {
		tmp = (1.0 + ((1.0 / (Math.sqrt((1.0 + z)) + Math.sqrt(z))) + t_1)) + (t_2 - Math.sqrt(y));
	} else {
		tmp = t_2 + ((1.0 / (t_1 + Math.sqrt(y))) + (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if z <= 1.5e+30:
		tmp = (1.0 + ((1.0 / (math.sqrt((1.0 + z)) + math.sqrt(z))) + t_1)) + (t_2 - math.sqrt(y))
	else:
		tmp = t_2 + ((1.0 / (t_1 + math.sqrt(y))) + (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (z <= 1.5e+30)
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + z)) + sqrt(z))) + t_1)) + Float64(t_2 - sqrt(y)));
	else
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (z <= 1.5e+30)
		tmp = (1.0 + ((1.0 / (sqrt((1.0 + z)) + sqrt(z))) + t_1)) + (t_2 - sqrt(y));
	else
		tmp = t_2 + ((1.0 / (t_1 + sqrt(y))) + (1.0 / (sqrt((1.0 + x)) + sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.5e+30], N[(N[(1.0 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.49999999999999989e30

   1. Initial program 94.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 94.6%

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

      2. associate-+r+ 94.6%

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

      3. associate-+r- 75.3%

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

      4. associate-+l- 63.1%

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

      5. associate-+r- 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in x around 0 28.3%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 28.3%

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

      2. associate--l+ 28.3%

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

   6. Simplified 28.3%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. flip-- 96.1%

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

      2. add-sqr-sqrt 93.9%

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

      3. add-sqr-sqrt 96.2%

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

   8. Applied egg-rr 28.5%

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right)\right) - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   9. Step-by-step derivation

      1. associate--l+ 98.0%

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

      2. +-inverses 98.0%

         \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1 + \color{blue}{0}}{\sqrt{1 + z} + \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 98.0%

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

   10. Simplified 28.5%

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

   if 1.49999999999999989e30 < z

   1. Initial program 87.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. +-commutative 87.6%

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

   3. Simplified 87.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)} \]
   4. Step-by-step derivation

      1. flip-- 87.5%

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

      2. add-sqr-sqrt 69.2%

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

      3. +-commutative 69.2%

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

      4. add-sqr-sqrt 87.7%

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

      5. +-commutative 87.7%

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

   5. Applied egg-rr 87.7%

      \[\leadsto \left(\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ 90.5%

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

      2. +-inverses 90.5%

         \[\leadsto \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

      3. metadata-eval 90.5%

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

   7. Simplified 90.5%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. flip-- 90.5%

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

      2. add-sqr-sqrt 72.8%

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

      3. add-sqr-sqrt 90.7%

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

   9. Applied egg-rr 90.7%

      \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 93.5%

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

       2. +-inverses 93.5%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

       3. metadata-eval 93.5%

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

   11. Simplified 93.5%

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

   12. Taylor expanded in z around inf 93.5%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. +-commutative 93.5%

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

   14. Simplified 93.5%

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\\ \end{array} \]

Alternative 9: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+22}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.8e+22)
   (+
    (+ 1.0 (+ (sqrt (+ 1.0 y)) (- (sqrt (+ 1.0 z)) (sqrt z))))
    (- (- (sqrt (+ 1.0 t)) (sqrt t)) (sqrt y)))
   (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.8e+22) {
		tmp = (1.0 + (sqrt((1.0 + y)) + (sqrt((1.0 + z)) - sqrt(z)))) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    if (y <= 6.8d+22) then
        tmp = (1.0d0 + (sqrt((1.0d0 + y)) + (sqrt((1.0d0 + z)) - sqrt(z)))) + ((sqrt((1.0d0 + t)) - sqrt(t)) - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.8e+22) {
		tmp = (1.0 + (Math.sqrt((1.0 + y)) + (Math.sqrt((1.0 + z)) - Math.sqrt(z)))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 6.8e+22:
		tmp = (1.0 + (math.sqrt((1.0 + y)) + (math.sqrt((1.0 + z)) - math.sqrt(z)))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.8e+22)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z)))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.8e+22)
		tmp = (1.0 + (sqrt((1.0 + y)) + (sqrt((1.0 + z)) - sqrt(z)))) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 6.8e+22], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8e22

   1. Initial program 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. associate-+r- 96.6%

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

      4. associate-+l- 96.6%

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

      5. associate-+r- 96.6%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in x around 0 32.5%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 52.9%

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

      2. associate--l+ 57.7%

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

   6. Simplified 57.7%

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

   if 6.8e22 < y

   1. Initial program 85.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.7%

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

      2. associate-+r+ 85.7%

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

      3. associate-+r- 50.1%

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

      4. associate-+l- 18.1%

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

      5. associate-+r- 3.2%

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

   3. Simplified 2.8%

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

   4. Taylor expanded in z around inf 2.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 2.7%

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

      2. associate-+r- 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in y around inf 18.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.3%

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

      2. associate--l+ 26.8%

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

      3. +-commutative 26.8%

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

   9. Simplified 26.8%

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

   10. Taylor expanded in t around inf 16.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 16.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 16.3%

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

       3. add-sqr-sqrt 16.0%

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

   12. Applied egg-rr 16.0%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 19.5%

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

       2. +-inverses 19.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 19.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 19.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 19.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+22}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 10: 89.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.45 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 1.45e-25)
     (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
     (if (<= y 5e+19)
       (+ t_1 (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ t_1 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 1.45e-25) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else if (y <= 5e+19) {
		tmp = t_1 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (t_1 + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 1.45d-25) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else if (y <= 5d+19) then
        tmp = t_1 + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (t_1 + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 1.45e-25) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 5e+19) {
		tmp = t_1 + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (t_1 + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 1.45e-25:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	elif y <= 5e+19:
		tmp = t_1 + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (t_1 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 1.45e-25)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	elseif (y <= 5e+19)
		tmp = Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 1.45e-25)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	elseif (y <= 5e+19)
		tmp = t_1 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (t_1 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.45e-25], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 5e+19], N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.45e-25

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. associate-+r- 97.1%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in x around 0 33.2%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 55.0%

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

      2. associate--l+ 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in y around 0 17.9%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 55.5%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 55.5%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 55.5%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 55.5%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in t around inf 55.7%

       \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 1.45e-25 < y < 5e19

   1. Initial program 94.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 94.0%

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

      2. associate-+r+ 94.0%

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

      3. associate-+r- 94.0%

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

      4. associate-+l- 94.0%

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

      5. associate-+r- 94.0%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in t around inf 6.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 13.8%

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

      2. associate--l+ 14.7%

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

      3. +-commutative 14.7%

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

   6. Simplified 14.7%

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

   7. Taylor expanded in z around inf 28.8%

      \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 28.8%

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

   9. Simplified 28.8%

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

   if 5e19 < y

   1. Initial program 85.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.6%

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

      2. associate-+r+ 85.6%

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

      3. associate-+r- 50.4%

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

      4. associate-+l- 18.6%

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

      5. associate-+r- 3.8%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in z around inf 2.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 2.8%

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

      2. associate-+r- 3.3%

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

   6. Simplified 3.3%

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

   7. Taylor expanded in y around inf 18.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.2%

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

      2. associate--l+ 26.7%

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

      3. +-commutative 26.7%

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

   9. Simplified 26.7%

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

   10. Taylor expanded in t around inf 16.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 16.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 16.3%

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

       3. add-sqr-sqrt 16.0%

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

   12. Applied egg-rr 16.0%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 19.5%

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

       2. +-inverses 19.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 19.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 19.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 19.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 11: 89.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.8e-26)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (if (<= y 4e+20)
     (exp (log1p (- (sqrt (+ 1.0 y)) (sqrt y))))
     (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.8e-26) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else if (y <= 4e+20) {
		tmp = exp(log1p((sqrt((1.0 + y)) - sqrt(y))));
	} else {
		tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.8e-26) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else if (y <= 4e+20) {
		tmp = Math.exp(Math.log1p((Math.sqrt((1.0 + y)) - Math.sqrt(y))));
	} else {
		tmp = 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 6.8e-26:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	elif y <= 4e+20:
		tmp = math.exp(math.log1p((math.sqrt((1.0 + y)) - math.sqrt(y))))
	else:
		tmp = 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.8e-26)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	elseif (y <= 4e+20)
		tmp = exp(log1p(Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 6.8e-26], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 4e+20], N[Exp[N[Log[1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.80000000000000026e-26

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. associate-+r- 97.1%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in x around 0 33.2%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 55.0%

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

      2. associate--l+ 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in y around 0 17.9%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 55.5%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 55.5%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 55.5%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 55.5%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in t around inf 55.7%

       \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 6.80000000000000026e-26 < y < 4e20

   1. Initial program 94.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 94.0%

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

      2. associate-+r+ 94.0%

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

      3. associate-+r- 94.0%

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

      4. associate-+l- 94.0%

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

      5. associate-+r- 94.0%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in x around 0 25.2%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 37.3%

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

      2. associate--l+ 57.4%

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

   6. Simplified 57.4%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

      1. add-exp-log 57.3%

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

      2. associate--l+ 57.3%

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

      3. log1p-def 57.3%

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

      4. associate--r- 42.6%

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

   8. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) - \left(\left(\sqrt{y} - \sqrt{1 + t}\right) + \sqrt{t}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate--l+ 42.6%

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

      2. associate-+l- 57.3%

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

      3. associate--r- 57.3%

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

   10. Simplified 57.3%

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

   11. Taylor expanded in t around inf 33.5%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)}\right)} \]

   12. Taylor expanded in z around inf 51.5%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{1 + y} - \sqrt{y}}\right)} \]

   if 4e20 < y

   1. Initial program 85.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.6%

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

      2. associate-+r+ 85.6%

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

      3. associate-+r- 50.4%

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

      4. associate-+l- 18.6%

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

      5. associate-+r- 3.8%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in z around inf 2.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 2.8%

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

      2. associate-+r- 3.3%

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

   6. Simplified 3.3%

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

   7. Taylor expanded in y around inf 18.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.2%

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

      2. associate--l+ 26.7%

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

      3. +-commutative 26.7%

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

   9. Simplified 26.7%

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

   10. Taylor expanded in t around inf 16.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 16.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 16.3%

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

       3. add-sqr-sqrt 16.0%

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

   12. Applied egg-rr 16.0%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 19.5%

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

       2. +-inverses 19.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 19.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 19.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 19.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-26}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\sqrt{1 + y} - \sqrt{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 12: 82.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.9)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (- (sqrt (+ 1.0 x)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.9) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    if (y <= 2.9d0) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.9) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.9:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.9)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.9)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 2.9], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.89999999999999991

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. associate-+r- 97.2%

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

      4. associate-+l- 97.2%

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

      5. associate-+r- 97.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in x around 0 32.2%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 54.0%

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

      2. associate--l+ 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in y around 0 17.2%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 53.9%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 53.9%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 53.9%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 53.9%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in t around inf 54.2%

       \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 2.89999999999999991 < y

   1. Initial program 85.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.7%

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

      2. associate-+r+ 85.7%

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

      3. associate-+r- 52.3%

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

      4. associate-+l- 22.2%

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

      5. associate-+r- 8.1%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in z around inf 4.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 4.3%

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

      2. associate-+r- 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in y around inf 17.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 17.7%

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

      2. associate--l+ 26.2%

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

      3. +-commutative 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in t around inf 16.1%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 13: 86.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.9)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) 2.0)
   (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.9) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	} else {
		tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    if (y <= 1.9d0) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + 2.0d0
    else
        tmp = 1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.9) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + 2.0;
	} else {
		tmp = 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.9:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + 2.0
	else:
		tmp = 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.9)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + 2.0);
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.9)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + 2.0;
	else
		tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.9], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8999999999999999

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.2%

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

      2. associate-+r+ 97.2%

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

      3. associate-+r- 97.2%

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

      4. associate-+l- 97.2%

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

      5. associate-+r- 97.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in x around 0 32.2%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 54.0%

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

      2. associate--l+ 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in y around 0 17.2%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 53.9%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 53.9%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 53.9%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 53.9%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in t around inf 54.2%

       \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} \]

   if 1.8999999999999999 < y

   1. Initial program 85.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.7%

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

      2. associate-+r+ 85.7%

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

      3. associate-+r- 52.3%

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

      4. associate-+l- 22.2%

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

      5. associate-+r- 8.1%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in z around inf 4.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 4.3%

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

      2. associate-+r- 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in y around inf 17.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 17.7%

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

      2. associate--l+ 26.2%

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

      3. +-commutative 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in t around inf 16.1%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 16.1%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 16.3%

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

       3. add-sqr-sqrt 16.1%

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

   12. Applied egg-rr 16.1%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \]
   13. Step-by-step derivation

       1. associate--l+ 19.5%

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

       2. +-inverses 19.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 19.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 19.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 19.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 14: 54.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4.8:\\ \;\;\;\;2 + \left(1 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.8) (+ 2.0 (- 1.0 (sqrt z))) (- (sqrt (+ 1.0 x)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.8) {
		tmp = 2.0 + (1.0 - sqrt(z));
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    if (z <= 4.8d0) then
        tmp = 2.0d0 + (1.0d0 - sqrt(z))
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.8) {
		tmp = 2.0 + (1.0 - Math.sqrt(z));
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 4.8:
		tmp = 2.0 + (1.0 - math.sqrt(z))
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.8)
		tmp = Float64(2.0 + Float64(1.0 - sqrt(z)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.8)
		tmp = 2.0 + (1.0 - sqrt(z));
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 4.8], N[(2.0 + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.79999999999999982

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.6%

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

      2. associate-+r+ 97.6%

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

      3. associate-+r- 78.5%

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

      4. associate-+l- 65.4%

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

      5. associate-+r- 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in x around 0 29.0%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.0%

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

      2. associate--l+ 29.0%

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

   6. Simplified 29.0%

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

   7. Taylor expanded in y around 0 21.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 45.0%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 45.0%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 45.0%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 45.0%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in z around 0 45.0%

       \[\leadsto 2 + \left(\color{blue}{\left(1 + \sqrt{1 + t}\right)} - \left(\sqrt{z} + \sqrt{t}\right)\right) \]

   11. Taylor expanded in t around inf 41.3%

       \[\leadsto 2 + \color{blue}{\left(1 - \sqrt{z}\right)} \]

   if 4.79999999999999982 < z

   1. Initial program 85.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.6%

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

      2. associate-+r+ 85.6%

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

      3. associate-+r- 69.5%

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

      4. associate-+l- 51.3%

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

      5. associate-+r- 51.3%

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

   3. Simplified 10.3%

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

   4. Taylor expanded in z around inf 31.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 31.7%

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

      2. associate-+r- 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in y around inf 18.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.3%

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

      2. associate--l+ 26.8%

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

      3. +-commutative 26.8%

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

   9. Simplified 26.8%

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

   10. Taylor expanded in t around inf 16.3%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8:\\ \;\;\;\;2 + \left(1 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 15: 53.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;2 + \left(1 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 4.0) (+ 2.0 (- 1.0 (sqrt z))) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.0) {
		tmp = 2.0 + (1.0 - sqrt(z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    if (z <= 4.0d0) then
        tmp = 2.0d0 + (1.0d0 - sqrt(z))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.0) {
		tmp = 2.0 + (1.0 - Math.sqrt(z));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 4.0:
		tmp = 2.0 + (1.0 - math.sqrt(z))
	else:
		tmp = 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.0)
		tmp = Float64(2.0 + Float64(1.0 - sqrt(z)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.0)
		tmp = 2.0 + (1.0 - sqrt(z));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 4.0], N[(2.0 + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if z < 4

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.6%

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

      2. associate-+r+ 97.6%

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

      3. associate-+r- 78.5%

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

      4. associate-+l- 65.4%

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

      5. associate-+r- 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in x around 0 29.0%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.0%

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

      2. associate--l+ 29.0%

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

   6. Simplified 29.0%

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

   7. Taylor expanded in y around 0 21.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 45.0%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 45.0%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 45.0%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 45.0%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in z around 0 45.0%

       \[\leadsto 2 + \left(\color{blue}{\left(1 + \sqrt{1 + t}\right)} - \left(\sqrt{z} + \sqrt{t}\right)\right) \]

   11. Taylor expanded in t around inf 41.3%

       \[\leadsto 2 + \color{blue}{\left(1 - \sqrt{z}\right)} \]

   if 4 < z

   1. Initial program 85.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.6%

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

      2. associate-+r+ 85.6%

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

      3. associate-+r- 69.5%

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

      4. associate-+l- 51.3%

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

      5. associate-+r- 51.3%

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

   3. Simplified 10.3%

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

   4. Taylor expanded in z around inf 31.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 31.7%

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

      2. associate-+r- 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in y around inf 18.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.3%

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

      2. associate--l+ 26.8%

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

      3. +-commutative 26.8%

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

   9. Simplified 26.8%

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

   10. Taylor expanded in t around inf 16.3%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 42.6%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;2 + \left(1 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 53.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;3 - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (if (<= z 4.0) (- 3.0 (sqrt z)) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.0) {
		tmp = 3.0 - sqrt(z);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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) :: tmp
    if (z <= 4.0d0) then
        tmp = 3.0d0 - sqrt(z)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 4.0) {
		tmp = 3.0 - Math.sqrt(z);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 4.0:
		tmp = 3.0 - math.sqrt(z)
	else:
		tmp = 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 4.0)
		tmp = Float64(3.0 - sqrt(z));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 4.0)
		tmp = 3.0 - sqrt(z);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 4.0], N[(3.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if z < 4

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 97.6%

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

      2. associate-+r+ 97.6%

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

      3. associate-+r- 78.5%

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

      4. associate-+l- 65.4%

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

      5. associate-+r- 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in x around 0 29.0%

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \sqrt{z}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. associate--l+ 29.0%

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

      2. associate--l+ 29.0%

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

   6. Simplified 29.0%

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

   7. Taylor expanded in y around 0 21.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 45.0%

         \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\sqrt{t} + \sqrt{z}\right)\right)} \]

      2. +-commutative 45.0%

         \[\leadsto 2 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \sqrt{z}\right)\right) \]

      3. +-commutative 45.0%

         \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \color{blue}{\left(\sqrt{z} + \sqrt{t}\right)}\right) \]

   9. Simplified 45.0%

      \[\leadsto \color{blue}{2 + \left(\left(\sqrt{1 + z} + \sqrt{1 + t}\right) - \left(\sqrt{z} + \sqrt{t}\right)\right)} \]

   10. Taylor expanded in z around 0 45.0%

       \[\leadsto 2 + \left(\color{blue}{\left(1 + \sqrt{1 + t}\right)} - \left(\sqrt{z} + \sqrt{t}\right)\right) \]

   11. Taylor expanded in t around inf 41.3%

       \[\leadsto \color{blue}{3 - \sqrt{z}} \]

   if 4 < z

   1. Initial program 85.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.6%

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

      2. associate-+r+ 85.6%

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

      3. associate-+r- 69.5%

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

      4. associate-+l- 51.3%

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

      5. associate-+r- 51.3%

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

   3. Simplified 10.3%

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

   4. Taylor expanded in z around inf 31.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 31.7%

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

      2. associate-+r- 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in y around inf 18.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 18.3%

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

      2. associate--l+ 26.8%

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

      3. +-commutative 26.8%

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

   9. Simplified 26.8%

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

   10. Taylor expanded in t around inf 16.3%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   11. Taylor expanded in x around 0 42.6%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;3 - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17: 34.8% accurate, 823.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 1.0)

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this 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 = 1.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return 1.0
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 91.2%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. +-commutative 91.2%

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

   2. associate-+r+ 91.2%

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

   3. associate-+r- 73.7%

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

   4. associate-+l- 57.9%

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

   5. associate-+r- 50.6%

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

3. Simplified 28.8%

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

4. Taylor expanded in z around inf 21.2%

   \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{x}\right)} - \left(\sqrt{y} - \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
5. Step-by-step derivation

   1. +-commutative 21.2%

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

   2. associate-+r- 32.4%

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

6. Simplified 32.4%

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

7. Taylor expanded in y around inf 13.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + t} + \sqrt{1 + x}\right) - \left(\sqrt{t} + \sqrt{x}\right)} \]
8. Step-by-step derivation

   1. +-commutative 13.1%

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

   2. associate--l+ 19.6%

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

   3. +-commutative 19.6%

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

9. Simplified 19.6%

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

10. Taylor expanded in t around inf 13.7%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Taylor expanded in x around 0 35.0%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 35.0%

    \[\leadsto 1 \]

Developer target: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 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))))

C

double code(double x, double y, double z, double t) {
	return (((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));
}

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 = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

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

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((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));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023322 
(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))))