Main:z from

Percentage Accurate: 92.0% → 99.9%

Time: 45.7s

Alternatives: 22

Speedup: 1.0×

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: 92.0% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))) + (((1.0d0 + (y - y)) / (sqrt((1.0d0 + y)) + sqrt(y))) + (((1.0d0 + (z - z)) / (sqrt((1.0d0 + z)) + sqrt(z))) + ((1.0d0 + (t - t)) / (sqrt((1.0d0 + t)) + sqrt(t)))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)))) + (((1.0 + (y - y)) / (sqrt((1.0 + y)) + sqrt(y))) + (((1.0 + (z - z)) / (sqrt((1.0 + z)) + sqrt(z))) + ((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t)))));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(t - t), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. +-commutative 91.4%

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

   5. +-commutative 91.4%

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

3. Simplified 91.4%

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

   1. flip-- 91.3%

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

   2. rem-square-sqrt 74.0%

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

   3. +-commutative 74.0%

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

   4. rem-square-sqrt 91.8%

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

   5. +-commutative 91.8%

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

   6. +-commutative 91.8%

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

5. Applied egg-rr 91.8%

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

   1. associate--l+ 94.1%

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

7. Simplified 94.1%

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

   1. flip-- 94.2%

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

   2. rem-square-sqrt 71.4%

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

   3. rem-square-sqrt 94.5%

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

   4. +-commutative 94.5%

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

9. Applied egg-rr 94.5%

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

    1. associate--l+ 96.0%

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

    2. +-commutative 96.0%

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

11. Simplified 96.0%

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

    1. flip-- 96.0%

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

    2. rem-square-sqrt 74.3%

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

    3. rem-square-sqrt 96.4%

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

13. Applied egg-rr 96.4%

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

    1. associate--l+ 97.5%

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

15. Simplified 97.5%

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

    1. flip-- 97.5%

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

    2. rem-square-sqrt 72.7%

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

    3. rem-square-sqrt 97.7%

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

17. Applied egg-rr 97.7%

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

    1. associate--l+ 99.8%

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

19. Simplified 99.8%

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

20. Final simplification 99.8%

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

Alternative 2: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := t_2 - \sqrt{z}\\ t_4 := \sqrt{1 + y}\\ \mathbf{if}\;t_3 \leq 0.0002:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_1} + \left(\frac{1}{t_2 + \sqrt{z}} + \frac{1}{t_4 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t_4 - \sqrt{y}\right) + \left(\frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} + t_3\right)\right) + \left(t_1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= t_3 0.0002)
     (+
      (/ (+ 1.0 (- x x)) (+ (sqrt x) t_1))
      (+ (/ 1.0 (+ t_2 (sqrt z))) (/ 1.0 (+ t_4 (sqrt y)))))
     (+
      (+
       (- t_4 (sqrt y))
       (+ (/ (+ 1.0 (- t t)) (+ (sqrt (+ 1.0 t)) (sqrt t))) t_3))
      (- t_1 (sqrt x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 0.0002) {
		tmp = ((1.0 + (x - x)) / (sqrt(x) + t_1)) + ((1.0 / (t_2 + sqrt(z))) + (1.0 / (t_4 + sqrt(y))));
	} else {
		tmp = ((t_4 - sqrt(y)) + (((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t))) + t_3)) + (t_1 - sqrt(x));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + y))
    if (t_3 <= 0.0002d0) then
        tmp = ((1.0d0 + (x - x)) / (sqrt(x) + t_1)) + ((1.0d0 / (t_2 + sqrt(z))) + (1.0d0 / (t_4 + sqrt(y))))
    else
        tmp = ((t_4 - sqrt(y)) + (((1.0d0 + (t - t)) / (sqrt((1.0d0 + t)) + sqrt(t))) + t_3)) + (t_1 - sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + y));
	double tmp;
	if (t_3 <= 0.0002) {
		tmp = ((1.0 + (x - x)) / (Math.sqrt(x) + t_1)) + ((1.0 / (t_2 + Math.sqrt(z))) + (1.0 / (t_4 + Math.sqrt(y))));
	} else {
		tmp = ((t_4 - Math.sqrt(y)) + (((1.0 + (t - t)) / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + t_3)) + (t_1 - Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + y))
	tmp = 0
	if t_3 <= 0.0002:
		tmp = ((1.0 + (x - x)) / (math.sqrt(x) + t_1)) + ((1.0 / (t_2 + math.sqrt(z))) + (1.0 / (t_4 + math.sqrt(y))))
	else:
		tmp = ((t_4 - math.sqrt(y)) + (((1.0 + (t - t)) / (math.sqrt((1.0 + t)) + math.sqrt(t))) + t_3)) + (t_1 - math.sqrt(x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_3 <= 0.0002)
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_1)) + Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(1.0 / Float64(t_4 + sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(t_4 - sqrt(y)) + Float64(Float64(Float64(1.0 + Float64(t - t)) / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + t_3)) + Float64(t_1 - sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t_3 <= 0.0002)
		tmp = ((1.0 + (x - x)) / (sqrt(x) + t_1)) + ((1.0 / (t_2 + sqrt(z))) + (1.0 / (t_4 + sqrt(y))));
	else
		tmp = ((t_4 - sqrt(y)) + (((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t))) + t_3)) + (t_1 - sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0002], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(t - t), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 2.0000000000000001e-4

   1. Initial program 86.0%

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

      1. associate-+l+ 86.0%

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

      2. associate-+l+ 86.0%

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

      3. +-commutative 86.0%

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

      4. +-commutative 86.0%

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

      5. +-commutative 86.0%

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

   3. Simplified 86.0%

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

      1. flip-- 86.0%

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

      2. rem-square-sqrt 71.6%

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

      3. +-commutative 71.6%

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

      4. rem-square-sqrt 86.4%

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

      5. +-commutative 86.4%

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

      6. +-commutative 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. associate--l+ 90.5%

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

   7. Simplified 90.5%

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

      1. flip-- 90.7%

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

      2. rem-square-sqrt 69.9%

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

      3. rem-square-sqrt 91.3%

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

      4. +-commutative 91.3%

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

   9. Applied egg-rr 91.3%

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

       1. associate--l+ 93.4%

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

       2. +-commutative 93.4%

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

   11. Simplified 93.4%

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

       1. flip-- 93.4%

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

       2. rem-square-sqrt 50.7%

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

       3. rem-square-sqrt 94.3%

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

   13. Applied egg-rr 94.3%

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

       1. associate--l+ 96.3%

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

   15. Simplified 96.3%

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

   16. Taylor expanded in t around inf 51.9%

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

       1. +-commutative 51.9%

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

   18. Simplified 51.9%

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

   if 2.0000000000000001e-4 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-+l+ 96.9%

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

      3. +-commutative 96.9%

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

      4. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

   3. Simplified 96.9%

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

      1. flip-- 98.7%

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

      2. rem-square-sqrt 71.6%

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

      3. rem-square-sqrt 99.1%

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

   5. Applied egg-rr 97.3%

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

      1. associate--l+ 99.9%

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

   7. Simplified 98.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0.0002:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \end{array} \]

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((1.0d0 + (x - x)) / (sqrt(x) + sqrt((1.0d0 + x)))) + (((1.0d0 + (y - y)) / (sqrt((1.0d0 + y)) + sqrt(y))) + (((1.0d0 + (z - z)) / (sqrt((1.0d0 + z)) + sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((1.0 + (x - x)) / (sqrt(x) + sqrt((1.0 + x)))) + (((1.0 + (y - y)) / (sqrt((1.0 + y)) + sqrt(y))) + (((1.0 + (z - z)) / (sqrt((1.0 + z)) + sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t))));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

   1. associate-+l+ 91.4%

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

   2. associate-+l+ 91.4%

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

   3. +-commutative 91.4%

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

   4. +-commutative 91.4%

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

   5. +-commutative 91.4%

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

3. Simplified 91.4%

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

   1. flip-- 91.3%

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

   2. rem-square-sqrt 74.0%

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

   3. +-commutative 74.0%

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

   4. rem-square-sqrt 91.8%

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

   5. +-commutative 91.8%

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

   6. +-commutative 91.8%

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

5. Applied egg-rr 91.8%

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

   1. associate--l+ 94.1%

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

7. Simplified 94.1%

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

   1. flip-- 94.2%

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

   2. rem-square-sqrt 71.4%

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

   3. rem-square-sqrt 94.5%

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

   4. +-commutative 94.5%

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

9. Applied egg-rr 94.5%

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

    1. associate--l+ 96.0%

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

    2. +-commutative 96.0%

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

11. Simplified 96.0%

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

    1. flip-- 96.0%

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

    2. rem-square-sqrt 74.3%

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

    3. rem-square-sqrt 96.4%

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

13. Applied egg-rr 96.4%

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

    1. associate--l+ 97.5%

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

15. Simplified 97.5%

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

16. Final simplification 97.5%

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

Alternative 4: 58.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z} + \sqrt{z}\\ t_3 := \sqrt{x} + t_1\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{t_3} + \left(\frac{1}{t_2} + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;\left(t_1 + \sqrt{1 + y}\right) + \left(\frac{1 + \left(z - z\right)}{t_2} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (+ (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (+ (sqrt x) t_1)))
   (if (<= y 4.2e-30)
     (+
      (/ (+ 1.0 (- x x)) t_3)
      (+ (/ 1.0 t_2) (+ 1.0 (- (sqrt (+ 1.0 t)) (sqrt t)))))
     (if (<= y 2.15e+23)
       (+
        (+ t_1 (sqrt (+ 1.0 y)))
        (- (/ (+ 1.0 (- z z)) t_2) (+ (sqrt x) (sqrt y))))
       (/ 1.0 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z)) + sqrt(z);
	double t_3 = sqrt(x) + t_1;
	double tmp;
	if (y <= 4.2e-30) {
		tmp = ((1.0 + (x - x)) / t_3) + ((1.0 / t_2) + (1.0 + (sqrt((1.0 + t)) - sqrt(t))));
	} else if (y <= 2.15e+23) {
		tmp = (t_1 + sqrt((1.0 + y))) + (((1.0 + (z - z)) / t_2) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / t_3;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z)) + sqrt(z)
    t_3 = sqrt(x) + t_1
    if (y <= 4.2d-30) then
        tmp = ((1.0d0 + (x - x)) / t_3) + ((1.0d0 / t_2) + (1.0d0 + (sqrt((1.0d0 + t)) - sqrt(t))))
    else if (y <= 2.15d+23) then
        tmp = (t_1 + sqrt((1.0d0 + y))) + (((1.0d0 + (z - z)) / t_2) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z)) + Math.sqrt(z);
	double t_3 = Math.sqrt(x) + t_1;
	double tmp;
	if (y <= 4.2e-30) {
		tmp = ((1.0 + (x - x)) / t_3) + ((1.0 / t_2) + (1.0 + (Math.sqrt((1.0 + t)) - Math.sqrt(t))));
	} else if (y <= 2.15e+23) {
		tmp = (t_1 + Math.sqrt((1.0 + y))) + (((1.0 + (z - z)) / t_2) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z)) + math.sqrt(z)
	t_3 = math.sqrt(x) + t_1
	tmp = 0
	if y <= 4.2e-30:
		tmp = ((1.0 + (x - x)) / t_3) + ((1.0 / t_2) + (1.0 + (math.sqrt((1.0 + t)) - math.sqrt(t))))
	elif y <= 2.15e+23:
		tmp = (t_1 + math.sqrt((1.0 + y))) + (((1.0 + (z - z)) / t_2) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(sqrt(Float64(1.0 + z)) + sqrt(z))
	t_3 = Float64(sqrt(x) + t_1)
	tmp = 0.0
	if (y <= 4.2e-30)
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / t_3) + Float64(Float64(1.0 / t_2) + Float64(1.0 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))));
	elseif (y <= 2.15e+23)
		tmp = Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) + Float64(Float64(Float64(1.0 + Float64(z - z)) / t_2) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z)) + sqrt(z);
	t_3 = sqrt(x) + t_1;
	tmp = 0.0;
	if (y <= 4.2e-30)
		tmp = ((1.0 + (x - x)) / t_3) + ((1.0 / t_2) + (1.0 + (sqrt((1.0 + t)) - sqrt(t))));
	elseif (y <= 2.15e+23)
		tmp = (t_1 + sqrt((1.0 + y))) + (((1.0 + (z - z)) / t_2) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[y, 4.2e-30], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(N[(1.0 / t$95$2), $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+23], N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / t$95$3), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2000000000000004e-30

   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. associate-+l+ 97.1%

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

      2. associate-+l+ 97.1%

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

      3. +-commutative 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 97.0%

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

      2. rem-square-sqrt 80.2%

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

      3. +-commutative 80.2%

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

      4. rem-square-sqrt 97.7%

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

      5. +-commutative 97.7%

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

      6. +-commutative 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. associate--l+ 98.3%

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

   7. Simplified 98.3%

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

      1. flip-- 98.3%

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

      2. rem-square-sqrt 98.3%

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

      3. rem-square-sqrt 98.3%

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

      4. +-commutative 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. associate--l+ 98.3%

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

       2. +-commutative 98.3%

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

   11. Simplified 98.3%

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

       1. flip-- 98.3%

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

       2. rem-square-sqrt 72.7%

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

       3. rem-square-sqrt 98.4%

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

   13. Applied egg-rr 98.4%

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

       1. associate--l+ 99.1%

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

   15. Simplified 99.1%

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

   16. Taylor expanded in y around 0 59.4%

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

       1. associate--l+ 81.1%

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

       2. +-commutative 81.1%

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

       3. metadata-eval 81.1%

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

       4. +-inverses 81.1%

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

       5. +-commutative 81.1%

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

       6. associate-+r- 99.0%

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

       7. +-commutative 99.0%

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

       8. associate-+r+ 99.0%

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

       9. +-inverses 99.0%

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

       10. metadata-eval 99.0%

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

       11. +-commutative 99.0%

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

   18. Simplified 99.0%

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

   if 4.2000000000000004e-30 < y < 2.1499999999999999e23

   1. Initial program 90.4%

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

      1. associate-+l+ 90.4%

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

      2. +-commutative 90.4%

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

      3. associate-+r- 42.5%

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

      4. associate-+l- 33.1%

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

      5. +-commutative 33.1%

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

      6. associate--l+ 33.1%

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

      7. +-commutative 33.1%

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

   3. Simplified 22.8%

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

      1. add-cbrt-cube_binary64 22.0%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]

   5. Applied rewrite-once 22.0%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]
   6. Step-by-step derivation

   7. Simplified 29.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{{\left(\left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)}^{3}}} \]

   8. Taylor expanded in t around inf 7.7%

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

      1. associate-+r+ 7.7%

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

      2. +-commutative 7.7%

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

      3. +-commutative 7.7%

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

      4. associate-+r+ 7.7%

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

      5. associate--l+ 12.2%

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

      6. +-commutative 12.2%

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

      7. +-commutative 12.2%

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

      8. associate--r+ 25.1%

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

      9. +-commutative 25.1%

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

   10. Simplified 25.1%

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

       1. flip-- 98.2%

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

       2. rem-square-sqrt 82.7%

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

       3. rem-square-sqrt 99.5%

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

   12. Applied egg-rr 25.1%

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

       1. associate--l+ 99.5%

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

   14. Simplified 25.1%

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

   if 2.1499999999999999e23 < y

   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. associate-+l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate-+l- 66.4%

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

      6. associate-+r- 56.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in t around inf 19.8%

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

   5. Taylor expanded in z around inf 18.6%

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

      1. +-commutative 18.6%

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

   7. Simplified 18.6%

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

   8. Taylor expanded in y around inf 18.0%

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

      1. flip-- 18.0%

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

      2. div-inv 18.0%

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

      3. +-commutative 18.0%

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

      4. +-commutative 18.0%

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

      5. rem-square-sqrt 18.5%

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

      6. +-commutative 18.5%

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

      7. rem-square-sqrt 18.0%

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

   10. Applied egg-rr 18.0%

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

       1. associate-*r/ 18.0%

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

       2. *-rgt-identity 18.0%

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

       3. associate--l+ 23.4%

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

       4. +-inverses 23.4%

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

       5. metadata-eval 23.4%

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

   12. Simplified 23.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 5: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;t \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\left(1 + \left(\frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} + \left(t_1 - \sqrt{z}\right)\right)\right) + \left(t_2 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + t_2} + \left(\frac{1}{t_1 + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= t 3.4e+26)
     (+
      (+
       1.0
       (+ (/ (+ 1.0 (- t t)) (+ (sqrt (+ 1.0 t)) (sqrt t))) (- t_1 (sqrt z))))
      (- t_2 (sqrt x)))
     (+
      (/ (+ 1.0 (- x x)) (+ (sqrt x) t_2))
      (+ (/ 1.0 (+ t_1 (sqrt z))) (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (t <= 3.4e+26) {
		tmp = (1.0 + (((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t))) + (t_1 - sqrt(z)))) + (t_2 - sqrt(x));
	} else {
		tmp = ((1.0 + (x - x)) / (sqrt(x) + t_2)) + ((1.0 / (t_1 + sqrt(z))) + (1.0 / (sqrt((1.0 + y)) + sqrt(y))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (t <= 3.4d+26) then
        tmp = (1.0d0 + (((1.0d0 + (t - t)) / (sqrt((1.0d0 + t)) + sqrt(t))) + (t_1 - sqrt(z)))) + (t_2 - sqrt(x))
    else
        tmp = ((1.0d0 + (x - x)) / (sqrt(x) + t_2)) + ((1.0d0 / (t_1 + sqrt(z))) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (t <= 3.4e+26) {
		tmp = (1.0 + (((1.0 + (t - t)) / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + (t_1 - Math.sqrt(z)))) + (t_2 - Math.sqrt(x));
	} else {
		tmp = ((1.0 + (x - x)) / (Math.sqrt(x) + t_2)) + ((1.0 / (t_1 + Math.sqrt(z))) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if t <= 3.4e+26:
		tmp = (1.0 + (((1.0 + (t - t)) / (math.sqrt((1.0 + t)) + math.sqrt(t))) + (t_1 - math.sqrt(z)))) + (t_2 - math.sqrt(x))
	else:
		tmp = ((1.0 + (x - x)) / (math.sqrt(x) + t_2)) + ((1.0 / (t_1 + math.sqrt(z))) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t <= 3.4e+26)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(t - t)) / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + Float64(t_1 - sqrt(z)))) + Float64(t_2 - sqrt(x)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + t_2)) + Float64(Float64(1.0 / Float64(t_1 + sqrt(z))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t <= 3.4e+26)
		tmp = (1.0 + (((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t))) + (t_1 - sqrt(z)))) + (t_2 - sqrt(x));
	else
		tmp = ((1.0 + (x - x)) / (sqrt(x) + t_2)) + ((1.0 / (t_1 + sqrt(z))) + (1.0 / (sqrt((1.0 + y)) + sqrt(y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3.4e+26], N[(N[(1.0 + N[(N[(N[(1.0 + N[(t - t), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$1 + 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]]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.4000000000000003e26

   1. Initial program 96.0%

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

      1. associate-+l+ 96.0%

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

      2. associate-+l+ 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

   3. Simplified 96.0%

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

      1. flip-- 98.2%

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

      2. rem-square-sqrt 97.8%

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

      3. rem-square-sqrt 98.6%

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

   5. Applied egg-rr 96.4%

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

      1. associate--l+ 99.8%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in y around 0 55.1%

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

   if 3.4000000000000003e26 < t

   1. Initial program 85.8%

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

      1. associate-+l+ 85.8%

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

      2. associate-+l+ 85.8%

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

      3. +-commutative 85.8%

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

      4. +-commutative 85.8%

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

      5. +-commutative 85.8%

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

   3. Simplified 85.8%

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

      1. flip-- 85.8%

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

      2. rem-square-sqrt 70.9%

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

      3. +-commutative 70.9%

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

      4. rem-square-sqrt 86.3%

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

      5. +-commutative 86.3%

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

      6. +-commutative 86.3%

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

   5. Applied egg-rr 86.3%

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

      1. associate--l+ 90.6%

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

   7. Simplified 90.6%

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

      1. flip-- 90.6%

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

      2. rem-square-sqrt 71.2%

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

      3. rem-square-sqrt 91.3%

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

      4. +-commutative 91.3%

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

   9. Applied egg-rr 91.3%

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

       1. associate--l+ 93.8%

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

       2. +-commutative 93.8%

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

   11. Simplified 93.8%

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

       1. flip-- 93.8%

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

       2. rem-square-sqrt 77.0%

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

       3. rem-square-sqrt 94.9%

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

   13. Applied egg-rr 94.9%

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

       1. associate--l+ 96.5%

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

   15. Simplified 96.5%

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

   16. Taylor expanded in t around inf 96.5%

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

       1. +-commutative 96.5%

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

   18. Simplified 96.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\left(1 + \left(\frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{1 + x}} + \left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\right)\\ \end{array} \]

Alternative 6: 31.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;\left(t_2 + \sqrt{1 + y}\right) + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.2e-215)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 2.15e+23)
       (+
        (+ t_2 (sqrt (+ 1.0 y)))
        (- (/ (+ 1.0 (- z z)) (+ t_1 (sqrt z))) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 2.15e+23) {
		tmp = (t_2 + sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + sqrt(z))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.2d-215) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 2.15d+23) then
        tmp = (t_2 + sqrt((1.0d0 + y))) + (((1.0d0 + (z - z)) / (t_1 + sqrt(z))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 2.15e+23) {
		tmp = (t_2 + Math.sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + Math.sqrt(z))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.2e-215:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 2.15e+23:
		tmp = (t_2 + math.sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + math.sqrt(z))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.2e-215)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 2.15e+23)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) + Float64(Float64(Float64(1.0 + Float64(z - z)) / Float64(t_1 + sqrt(z))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.2e-215)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 2.15e+23)
		tmp = (t_2 + sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + sqrt(z))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.2e-215], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+23], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2e-215

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. associate-+l- 61.4%

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

      6. associate-+r- 57.9%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. associate--l+ 51.0%

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

      2. associate--l+ 52.6%

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

      3. associate-+r+ 52.6%

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

      4. +-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate--l+ 64.9%

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

      3. associate--l+ 61.2%

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

      4. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.2e-215 < y < 2.1499999999999999e23

   1. Initial program 96.4%

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

      1. associate-+l+ 96.4%

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

      2. +-commutative 96.4%

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

      3. associate-+r- 53.3%

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

      4. associate-+l- 46.2%

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

      5. +-commutative 46.2%

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

      6. associate--l+ 46.2%

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

      7. +-commutative 46.2%

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

   3. Simplified 32.1%

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

      1. add-cbrt-cube_binary64 31.6%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]

   5. Applied rewrite-once 31.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]
   6. Step-by-step derivation

   7. Simplified 38.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{{\left(\left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)}^{3}}} \]

   8. Taylor expanded in t around inf 14.6%

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

      1. associate-+r+ 14.6%

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

      2. +-commutative 14.6%

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

      3. +-commutative 14.6%

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

      4. associate-+r+ 14.6%

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

      5. associate--l+ 25.8%

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

      6. +-commutative 25.8%

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

      7. +-commutative 25.8%

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

      8. associate--r+ 28.8%

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

      9. +-commutative 28.8%

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

   10. Simplified 28.8%

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

       1. flip-- 98.7%

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

       2. rem-square-sqrt 76.5%

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

       3. rem-square-sqrt 99.1%

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

   12. Applied egg-rr 29.0%

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

       1. associate--l+ 99.1%

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

   14. Simplified 29.0%

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

   if 2.1499999999999999e23 < y

   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. associate-+l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate-+l- 66.4%

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

      6. associate-+r- 56.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in t around inf 19.8%

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

   5. Taylor expanded in z around inf 18.6%

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

      1. +-commutative 18.6%

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

   7. Simplified 18.6%

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

   8. Taylor expanded in y around inf 18.0%

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

      1. flip-- 18.0%

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

      2. div-inv 18.0%

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

      3. +-commutative 18.0%

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

      4. +-commutative 18.0%

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

      5. rem-square-sqrt 18.5%

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

      6. +-commutative 18.5%

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

      7. rem-square-sqrt 18.0%

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

   10. Applied egg-rr 18.0%

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

       1. associate-*r/ 18.0%

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

       2. *-rgt-identity 18.0%

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

       3. associate--l+ 23.4%

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

       4. +-inverses 23.4%

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

       5. metadata-eval 23.4%

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

   12. Simplified 23.4%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 7: 48.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.6 \cdot 10^{-127}:\\ \;\;\;\;\left(1 + \left(\frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} + \left(t_1 - \sqrt{z}\right)\right)\right) + \left(t_2 - \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;\left(t_2 + \sqrt{1 + y}\right) + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.6e-127)
     (+
      (+
       1.0
       (+ (/ (+ 1.0 (- t t)) (+ (sqrt (+ 1.0 t)) (sqrt t))) (- t_1 (sqrt z))))
      (- t_2 (sqrt x)))
     (if (<= y 2.15e+23)
       (+
        (+ t_2 (sqrt (+ 1.0 y)))
        (- (/ (+ 1.0 (- z z)) (+ t_1 (sqrt z))) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.6e-127) {
		tmp = (1.0 + (((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t))) + (t_1 - sqrt(z)))) + (t_2 - sqrt(x));
	} else if (y <= 2.15e+23) {
		tmp = (t_2 + sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + sqrt(z))) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.6d-127) then
        tmp = (1.0d0 + (((1.0d0 + (t - t)) / (sqrt((1.0d0 + t)) + sqrt(t))) + (t_1 - sqrt(z)))) + (t_2 - sqrt(x))
    else if (y <= 2.15d+23) then
        tmp = (t_2 + sqrt((1.0d0 + y))) + (((1.0d0 + (z - z)) / (t_1 + sqrt(z))) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.6e-127) {
		tmp = (1.0 + (((1.0 + (t - t)) / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) + (t_1 - Math.sqrt(z)))) + (t_2 - Math.sqrt(x));
	} else if (y <= 2.15e+23) {
		tmp = (t_2 + Math.sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + Math.sqrt(z))) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.6e-127:
		tmp = (1.0 + (((1.0 + (t - t)) / (math.sqrt((1.0 + t)) + math.sqrt(t))) + (t_1 - math.sqrt(z)))) + (t_2 - math.sqrt(x))
	elif y <= 2.15e+23:
		tmp = (t_2 + math.sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + math.sqrt(z))) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.6e-127)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(t - t)) / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) + Float64(t_1 - sqrt(z)))) + Float64(t_2 - sqrt(x)));
	elseif (y <= 2.15e+23)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) + Float64(Float64(Float64(1.0 + Float64(z - z)) / Float64(t_1 + sqrt(z))) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.6e-127)
		tmp = (1.0 + (((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t))) + (t_1 - sqrt(z)))) + (t_2 - sqrt(x));
	elseif (y <= 2.15e+23)
		tmp = (t_2 + sqrt((1.0 + y))) + (((1.0 + (z - z)) / (t_1 + sqrt(z))) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.6e-127], N[(N[(1.0 + N[(N[(N[(1.0 + N[(t - t), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+23], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.60000000000000038e-127

   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. associate-+l+ 97.1%

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

      2. associate-+l+ 97.1%

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

      3. +-commutative 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. flip-- 99.5%

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

      2. rem-square-sqrt 79.0%

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

      3. rem-square-sqrt 99.5%

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

   5. Applied egg-rr 97.1%

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

      1. associate--l+ 99.9%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in y around 0 97.5%

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

   if 4.60000000000000038e-127 < y < 2.1499999999999999e23

   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. associate-+l+ 94.6%

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

      2. +-commutative 94.6%

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

      3. associate-+r- 47.1%

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

      4. associate-+l- 39.3%

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

      5. +-commutative 39.3%

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

      6. associate--l+ 39.3%

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

      7. +-commutative 39.3%

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

   3. Simplified 24.3%

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

      1. add-cbrt-cube_binary64 23.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]

   5. Applied rewrite-once 23.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]
   6. Step-by-step derivation

   7. Simplified 30.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{{\left(\left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)}^{3}}} \]

   8. Taylor expanded in t around inf 14.3%

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

      1. associate-+r+ 14.3%

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

      2. +-commutative 14.3%

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

      3. +-commutative 14.3%

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

      4. associate-+r+ 14.3%

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

      5. associate--l+ 23.5%

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

      6. +-commutative 23.5%

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

      7. +-commutative 23.5%

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

      8. associate--r+ 28.2%

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

      9. +-commutative 28.2%

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

   10. Simplified 28.2%

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

       1. flip-- 97.9%

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

       2. rem-square-sqrt 76.5%

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

       3. rem-square-sqrt 98.7%

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

   12. Applied egg-rr 28.5%

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

       1. associate--l+ 98.7%

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

   14. Simplified 28.5%

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

   if 2.1499999999999999e23 < y

   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. associate-+l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate-+l- 66.4%

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

      6. associate-+r- 56.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in t around inf 19.8%

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

   5. Taylor expanded in z around inf 18.6%

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

      1. +-commutative 18.6%

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

   7. Simplified 18.6%

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

   8. Taylor expanded in y around inf 18.0%

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

      1. flip-- 18.0%

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

      2. div-inv 18.0%

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

      3. +-commutative 18.0%

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

      4. +-commutative 18.0%

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

      5. rem-square-sqrt 18.5%

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

      6. +-commutative 18.5%

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

      7. rem-square-sqrt 18.0%

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

   10. Applied egg-rr 18.0%

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

       1. associate-*r/ 18.0%

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

       2. *-rgt-identity 18.0%

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

       3. associate--l+ 23.4%

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

       4. +-inverses 23.4%

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

       5. metadata-eval 23.4%

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

   12. Simplified 23.4%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-127}:\\ \;\;\;\;\left(1 + \left(\frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 8: 31.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+18}:\\ \;\;\;\;\left(t_2 + \sqrt{1 + y}\right) + \left(\left(t_1 - \sqrt{z}\right) + \frac{x - y}{\sqrt{y} - \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.2e-215)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 2.5e+18)
       (+
        (+ t_2 (sqrt (+ 1.0 y)))
        (+ (- t_1 (sqrt z)) (/ (- x y) (- (sqrt y) (sqrt x)))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 2.5e+18) {
		tmp = (t_2 + sqrt((1.0 + y))) + ((t_1 - sqrt(z)) + ((x - y) / (sqrt(y) - sqrt(x))));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.2d-215) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 2.5d+18) then
        tmp = (t_2 + sqrt((1.0d0 + y))) + ((t_1 - sqrt(z)) + ((x - y) / (sqrt(y) - sqrt(x))))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 2.5e+18) {
		tmp = (t_2 + Math.sqrt((1.0 + y))) + ((t_1 - Math.sqrt(z)) + ((x - y) / (Math.sqrt(y) - Math.sqrt(x))));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.2e-215:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 2.5e+18:
		tmp = (t_2 + math.sqrt((1.0 + y))) + ((t_1 - math.sqrt(z)) + ((x - y) / (math.sqrt(y) - math.sqrt(x))))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.2e-215)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 2.5e+18)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) + Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(x - y) / Float64(sqrt(y) - sqrt(x)))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.2e-215)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 2.5e+18)
		tmp = (t_2 + sqrt((1.0 + y))) + ((t_1 - sqrt(z)) + ((x - y) / (sqrt(y) - sqrt(x))));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.2e-215], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+18], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(x - y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2e-215

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. associate-+l- 61.4%

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

      6. associate-+r- 57.9%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. associate--l+ 51.0%

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

      2. associate--l+ 52.6%

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

      3. associate-+r+ 52.6%

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

      4. +-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate--l+ 64.9%

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

      3. associate--l+ 61.2%

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

      4. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.2e-215 < y < 2.5e18

   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. associate-+l+ 96.6%

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

      2. +-commutative 96.6%

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

      3. associate-+r- 53.0%

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

      4. associate-+l- 45.8%

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

      5. +-commutative 45.8%

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

      6. associate--l+ 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 32.3%

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

      1. add-cbrt-cube_binary64 31.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]

   5. Applied rewrite-once 31.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]
   6. Step-by-step derivation

   7. Simplified 37.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{{\left(\left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)}^{3}}} \]

   8. Taylor expanded in t around inf 14.7%

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

      1. associate-+r+ 14.7%

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

      2. +-commutative 14.7%

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

      3. +-commutative 14.7%

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

      4. associate-+r+ 14.7%

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

      5. associate--l+ 25.9%

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

      6. +-commutative 25.9%

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

      7. +-commutative 25.9%

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

      8. associate--r+ 28.2%

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

      9. +-commutative 28.2%

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

   10. Simplified 28.2%

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

       1. flip-+ 20.2%

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

       2. div-sub 20.2%

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

       3. rem-square-sqrt 20.2%

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

       4. rem-square-sqrt 20.3%

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

   12. Applied egg-rr 27.6%

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

       1. div-sub 20.3%

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

   14. Simplified 27.6%

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

   if 2.5e18 < y

   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. associate-+l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate-+l- 66.5%

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

      6. associate-+r- 56.3%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in t around inf 20.3%

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

   5. Taylor expanded in z around inf 19.1%

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

      1. +-commutative 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. flip-- 18.5%

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

      2. div-inv 18.5%

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

      3. +-commutative 18.5%

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

      4. +-commutative 18.5%

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

      5. rem-square-sqrt 19.0%

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

      6. +-commutative 19.0%

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

      7. rem-square-sqrt 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. associate-*r/ 18.5%

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

       2. *-rgt-identity 18.5%

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

       3. associate--l+ 23.8%

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

       4. +-inverses 23.8%

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

       5. metadata-eval 23.8%

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

   12. Simplified 23.8%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{x - y}{\sqrt{y} - \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 9: 31.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.2e-215)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 1.7e+41)
       (+ t_2 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- (- t_1 (sqrt z)) (sqrt x))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 1.7e+41) {
		tmp = t_2 + ((sqrt((1.0 + y)) - sqrt(y)) + ((t_1 - sqrt(z)) - sqrt(x)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.2d-215) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 1.7d+41) then
        tmp = t_2 + ((sqrt((1.0d0 + y)) - sqrt(y)) + ((t_1 - sqrt(z)) - sqrt(x)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 1.7e+41) {
		tmp = t_2 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) + ((t_1 - Math.sqrt(z)) - Math.sqrt(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.2e-215:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 1.7e+41:
		tmp = t_2 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) + ((t_1 - math.sqrt(z)) - math.sqrt(x)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.2e-215)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 1.7e+41)
		tmp = Float64(t_2 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(Float64(t_1 - sqrt(z)) - sqrt(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.2e-215)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 1.7e+41)
		tmp = t_2 + ((sqrt((1.0 + y)) - sqrt(y)) + ((t_1 - sqrt(z)) - sqrt(x)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.2e-215], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+41], N[(t$95$2 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2e-215

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. associate-+l- 61.4%

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

      6. associate-+r- 57.9%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. associate--l+ 51.0%

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

      2. associate--l+ 52.6%

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

      3. associate-+r+ 52.6%

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

      4. +-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate--l+ 64.9%

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

      3. associate--l+ 61.2%

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

      4. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.2e-215 < y < 1.69999999999999999e41

   1. Initial program 95.4%

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

      1. associate-+l+ 95.4%

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

      2. +-commutative 95.4%

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

      3. associate-+r- 58.0%

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

      4. associate-+l- 47.5%

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

      5. +-commutative 47.5%

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

      6. associate--l+ 47.5%

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

      7. +-commutative 47.5%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around inf 29.0%

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

      1. associate--l+ 28.7%

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

   6. Simplified 28.7%

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

   if 1.69999999999999999e41 < y

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+l+ 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l- 66.2%

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

      6. associate-+r- 55.6%

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

   3. Simplified 25.1%

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

   4. Taylor expanded in t around inf 18.9%

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

   5. Taylor expanded in z around inf 18.8%

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

      1. +-commutative 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in y around inf 18.2%

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

      1. flip-- 18.2%

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

      2. div-inv 18.2%

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

      3. +-commutative 18.2%

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

      4. +-commutative 18.2%

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

      5. rem-square-sqrt 18.6%

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

      6. +-commutative 18.6%

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

      7. rem-square-sqrt 18.2%

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

   10. Applied egg-rr 18.2%

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

       1. associate-*r/ 18.2%

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

       2. *-rgt-identity 18.2%

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

       3. associate--l+ 24.0%

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

       4. +-inverses 24.0%

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

       5. metadata-eval 24.0%

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

   12. Simplified 24.0%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 10: 31.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;\left(t_2 + \sqrt{1 + y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.8e-215)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 4.4e+18)
       (+ (+ t_2 (sqrt (+ 1.0 y))) (- (- t_1 (sqrt z)) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.8e-215) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 4.4e+18) {
		tmp = (t_2 + sqrt((1.0 + y))) + ((t_1 - sqrt(z)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.8d-215) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 4.4d+18) then
        tmp = (t_2 + sqrt((1.0d0 + y))) + ((t_1 - sqrt(z)) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.8e-215) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 4.4e+18) {
		tmp = (t_2 + Math.sqrt((1.0 + y))) + ((t_1 - Math.sqrt(z)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.8e-215:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 4.4e+18:
		tmp = (t_2 + math.sqrt((1.0 + y))) + ((t_1 - math.sqrt(z)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.8e-215)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 4.4e+18)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) + Float64(Float64(t_1 - sqrt(z)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.8e-215)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 4.4e+18)
		tmp = (t_2 + sqrt((1.0 + y))) + ((t_1 - sqrt(z)) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.8e-215], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+18], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.8000000000000002e-215

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. associate-+l- 61.4%

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

      6. associate-+r- 57.9%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. associate--l+ 51.0%

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

      2. associate--l+ 52.6%

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

      3. associate-+r+ 52.6%

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

      4. +-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate--l+ 64.9%

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

      3. associate--l+ 61.2%

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

      4. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.8000000000000002e-215 < y < 4.4e18

   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. associate-+l+ 96.6%

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

      2. +-commutative 96.6%

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

      3. associate-+r- 53.0%

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

      4. associate-+l- 45.8%

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

      5. +-commutative 45.8%

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

      6. associate--l+ 45.8%

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

      7. +-commutative 45.8%

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

   3. Simplified 32.3%

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

      1. add-cbrt-cube_binary64 31.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]

   5. Applied rewrite-once 31.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]
   6. Step-by-step derivation

   7. Simplified 37.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{{\left(\left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)}^{3}}} \]

   8. Taylor expanded in t around inf 14.7%

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

      1. associate-+r+ 14.7%

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

      2. +-commutative 14.7%

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

      3. +-commutative 14.7%

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

      4. associate-+r+ 14.7%

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

      5. associate--l+ 25.9%

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

      6. +-commutative 25.9%

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

      7. +-commutative 25.9%

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

      8. associate--r+ 28.2%

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

      9. +-commutative 28.2%

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

   10. Simplified 28.2%

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

   if 4.4e18 < y

   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. associate-+l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate-+l- 66.5%

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

      6. associate-+r- 56.3%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in t around inf 20.3%

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

   5. Taylor expanded in z around inf 19.1%

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

      1. +-commutative 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. flip-- 18.5%

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

      2. div-inv 18.5%

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

      3. +-commutative 18.5%

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

      4. +-commutative 18.5%

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

      5. rem-square-sqrt 19.0%

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

      6. +-commutative 19.0%

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

      7. rem-square-sqrt 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. associate-*r/ 18.5%

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

       2. *-rgt-identity 18.5%

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

       3. associate--l+ 23.8%

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

       4. +-inverses 23.8%

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

       5. metadata-eval 23.8%

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

   12. Simplified 23.8%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 11: 31.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 1.7 \cdot 10^{-214}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 42000000000:\\ \;\;\;\;\left(t_2 + \sqrt{1 + y}\right) + \left(\left(t_1 - \sqrt{z}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 1.7e-214)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 42000000000.0)
       (+ (+ t_2 (sqrt (+ 1.0 y))) (- (- t_1 (sqrt z)) (sqrt y)))
       (/ 1.0 (+ (sqrt x) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 1.7e-214) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 42000000000.0) {
		tmp = (t_2 + sqrt((1.0 + y))) + ((t_1 - sqrt(z)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 1.7d-214) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 42000000000.0d0) then
        tmp = (t_2 + sqrt((1.0d0 + y))) + ((t_1 - sqrt(z)) - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 1.7e-214) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 42000000000.0) {
		tmp = (t_2 + Math.sqrt((1.0 + y))) + ((t_1 - Math.sqrt(z)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 1.7e-214:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 42000000000.0:
		tmp = (t_2 + math.sqrt((1.0 + y))) + ((t_1 - math.sqrt(z)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 1.7e-214)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 42000000000.0)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) + Float64(Float64(t_1 - sqrt(z)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 1.7e-214)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 42000000000.0)
		tmp = (t_2 + sqrt((1.0 + y))) + ((t_1 - sqrt(z)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.7e-214], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 42000000000.0], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.7e-214

   1. Initial program 95.2%

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

      1. associate-+l+ 95.2%

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

      2. +-commutative 95.2%

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

      3. associate-+l+ 95.2%

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

      4. +-commutative 95.2%

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

      5. associate-+l- 62.3%

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

      6. associate-+r- 58.9%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in x around 0 29.7%

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

      1. associate--l+ 50.2%

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

      2. associate--l+ 51.8%

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

      3. associate-+r+ 51.8%

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

      4. +-commutative 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in y around 0 29.7%

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

      1. +-commutative 29.7%

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

      2. associate--l+ 63.8%

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

      3. associate--l+ 62.1%

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

      4. +-commutative 62.1%

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

   9. Simplified 62.1%

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

   if 1.7e-214 < y < 4.2e10

   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. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. associate-+r- 52.1%

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

      4. associate-+l- 44.7%

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

      5. +-commutative 44.7%

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

      6. associate--l+ 44.7%

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

      7. +-commutative 44.7%

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

   3. Simplified 30.7%

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

      1. add-cbrt-cube_binary64 30.2%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]

   5. Applied rewrite-once 30.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{\left(\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)\right) \cdot \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \sqrt{1 + t}\right) + \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \sqrt{z}\right)\right)\right)}} \]
   6. Step-by-step derivation

   7. Simplified 36.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\sqrt[3]{{\left(\left(\sqrt{1 + t} - \left(\sqrt{t} - \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)}^{3}}} \]

   8. Taylor expanded in t around inf 14.9%

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

      1. associate-+r+ 14.9%

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

      2. +-commutative 14.9%

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

      3. +-commutative 14.9%

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

      4. associate-+r+ 14.9%

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

      5. associate--l+ 26.2%

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

      6. +-commutative 26.2%

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

      7. +-commutative 26.2%

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

      8. associate--r+ 28.5%

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

      9. +-commutative 28.5%

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

   10. Simplified 28.5%

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

   11. Taylor expanded in x around 0 28.5%

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

   if 4.2e10 < y

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+l+ 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l- 66.5%

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

      6. associate-+r- 56.5%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in t around inf 20.3%

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

   5. Taylor expanded in z around inf 19.1%

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

      1. +-commutative 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. flip-- 18.5%

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

      2. div-inv 18.5%

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

      3. +-commutative 18.5%

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

      4. +-commutative 18.5%

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

      5. rem-square-sqrt 18.9%

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

      6. +-commutative 18.9%

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

      7. rem-square-sqrt 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. associate-*r/ 18.5%

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

       2. *-rgt-identity 18.5%

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

       3. associate--l+ 23.7%

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

       4. +-inverses 23.7%

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

       5. metadata-eval 23.7%

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

   12. Simplified 23.7%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-214}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 42000000000:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 12: 38.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.3 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;t_2 + \left(\sqrt{1 + y} - \frac{y - x}{\sqrt{y} - \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.3e-215)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 1.65e-23)
       (+ 2.0 (- t_1 (+ (sqrt y) (sqrt z))))
       (if (<= y 3.6e+15)
         (+ t_2 (- (sqrt (+ 1.0 y)) (/ (- y x) (- (sqrt y) (sqrt x)))))
         (/ 1.0 (+ (sqrt x) t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.3e-215) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 1.65e-23) {
		tmp = 2.0 + (t_1 - (sqrt(y) + sqrt(z)));
	} else if (y <= 3.6e+15) {
		tmp = t_2 + (sqrt((1.0 + y)) - ((y - x) / (sqrt(y) - sqrt(x))));
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.3d-215) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 1.65d-23) then
        tmp = 2.0d0 + (t_1 - (sqrt(y) + sqrt(z)))
    else if (y <= 3.6d+15) then
        tmp = t_2 + (sqrt((1.0d0 + y)) - ((y - x) / (sqrt(y) - sqrt(x))))
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.3e-215) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 1.65e-23) {
		tmp = 2.0 + (t_1 - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (y <= 3.6e+15) {
		tmp = t_2 + (Math.sqrt((1.0 + y)) - ((y - x) / (Math.sqrt(y) - Math.sqrt(x))));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.3e-215:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 1.65e-23:
		tmp = 2.0 + (t_1 - (math.sqrt(y) + math.sqrt(z)))
	elif y <= 3.6e+15:
		tmp = t_2 + (math.sqrt((1.0 + y)) - ((y - x) / (math.sqrt(y) - math.sqrt(x))))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.3e-215)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 1.65e-23)
		tmp = Float64(2.0 + Float64(t_1 - Float64(sqrt(y) + sqrt(z))));
	elseif (y <= 3.6e+15)
		tmp = Float64(t_2 + Float64(sqrt(Float64(1.0 + y)) - Float64(Float64(y - x) / Float64(sqrt(y) - sqrt(x)))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.3e-215)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 1.65e-23)
		tmp = 2.0 + (t_1 - (sqrt(y) + sqrt(z)));
	elseif (y <= 3.6e+15)
		tmp = t_2 + (sqrt((1.0 + y)) - ((y - x) / (sqrt(y) - sqrt(x))));
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.3e-215], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e-23], N[(2.0 + N[(t$95$1 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+15], N[(t$95$2 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[(y - x), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.30000000000000024e-215

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. associate-+l- 61.4%

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

      6. associate-+r- 57.9%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. associate--l+ 51.0%

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

      2. associate--l+ 52.6%

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

      3. associate-+r+ 52.6%

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

      4. +-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate--l+ 64.9%

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

      3. associate--l+ 61.2%

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

      4. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.30000000000000024e-215 < y < 1.6500000000000001e-23

   1. Initial program 98.3%

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

      1. associate-+l+ 98.3%

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

      2. +-commutative 98.3%

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

      3. associate-+l+ 98.3%

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

      4. +-commutative 98.3%

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

      5. associate-+l- 54.4%

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

      6. associate-+r- 51.1%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in t around inf 21.3%

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

   5. Taylor expanded in y around 0 21.3%

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

   6. Taylor expanded in x around 0 27.9%

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

      1. associate--l+ 54.7%

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

      2. +-commutative 54.7%

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

   8. Simplified 54.7%

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

   if 1.6500000000000001e-23 < y < 3.6e15

   1. Initial program 91.1%

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

      1. associate-+l+ 91.1%

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

      2. +-commutative 91.1%

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

      3. associate-+l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate-+l- 50.5%

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

      6. associate-+r- 46.4%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in t around inf 9.8%

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

   5. Taylor expanded in z around inf 21.2%

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

      1. +-commutative 21.2%

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

   7. Simplified 21.2%

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

      1. flip-+ 21.1%

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

      2. div-sub 21.1%

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

      3. rem-square-sqrt 21.2%

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

      4. rem-square-sqrt 21.9%

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

   9. Applied egg-rr 21.9%

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

       1. div-sub 21.9%

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

   11. Simplified 21.9%

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

   if 3.6e15 < y

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+l+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+l- 66.4%

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

      6. associate-+r- 56.3%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in t around inf 20.3%

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

   5. Taylor expanded in z around inf 19.1%

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

      1. +-commutative 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. flip-- 18.5%

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

      2. div-inv 18.5%

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

      3. +-commutative 18.5%

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

      4. +-commutative 18.5%

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

      5. rem-square-sqrt 18.9%

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

      6. +-commutative 18.9%

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

      7. rem-square-sqrt 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. associate-*r/ 18.5%

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

       2. *-rgt-identity 18.5%

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

       3. associate--l+ 23.8%

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

       4. +-inverses 23.8%

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

       5. metadata-eval 23.8%

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

   12. Simplified 23.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \frac{y - x}{\sqrt{y} - \sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 13: 37.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(t_1 - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;2 + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(t_2 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 4.2e-215)
     (+ 2.0 (+ (sqrt (+ 1.0 t)) (- t_1 (+ (sqrt z) (sqrt t)))))
     (if (<= y 2.05e-35)
       (+ 2.0 (- t_1 (+ (sqrt y) (sqrt z))))
       (if (<= y 4.5e+15)
         (- (- (+ t_2 (sqrt (+ 1.0 y))) (sqrt y)) (sqrt x))
         (/ 1.0 (+ (sqrt x) t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	} else if (y <= 2.05e-35) {
		tmp = 2.0 + (t_1 - (sqrt(y) + sqrt(z)));
	} else if (y <= 4.5e+15) {
		tmp = ((t_2 + sqrt((1.0 + y))) - sqrt(y)) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt(x) + t_2);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = sqrt((1.0d0 + x))
    if (y <= 4.2d-215) then
        tmp = 2.0d0 + (sqrt((1.0d0 + t)) + (t_1 - (sqrt(z) + sqrt(t))))
    else if (y <= 2.05d-35) then
        tmp = 2.0d0 + (t_1 - (sqrt(y) + sqrt(z)))
    else if (y <= 4.5d+15) then
        tmp = ((t_2 + sqrt((1.0d0 + y))) - sqrt(y)) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt(x) + t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 4.2e-215) {
		tmp = 2.0 + (Math.sqrt((1.0 + t)) + (t_1 - (Math.sqrt(z) + Math.sqrt(t))));
	} else if (y <= 2.05e-35) {
		tmp = 2.0 + (t_1 - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (y <= 4.5e+15) {
		tmp = ((t_2 + Math.sqrt((1.0 + y))) - Math.sqrt(y)) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 4.2e-215:
		tmp = 2.0 + (math.sqrt((1.0 + t)) + (t_1 - (math.sqrt(z) + math.sqrt(t))))
	elif y <= 2.05e-35:
		tmp = 2.0 + (t_1 - (math.sqrt(y) + math.sqrt(z)))
	elif y <= 4.5e+15:
		tmp = ((t_2 + math.sqrt((1.0 + y))) - math.sqrt(y)) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt(x) + t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 4.2e-215)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + t)) + Float64(t_1 - Float64(sqrt(z) + sqrt(t)))));
	elseif (y <= 2.05e-35)
		tmp = Float64(2.0 + Float64(t_1 - Float64(sqrt(y) + sqrt(z))));
	elseif (y <= 4.5e+15)
		tmp = Float64(Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) - sqrt(y)) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 4.2e-215)
		tmp = 2.0 + (sqrt((1.0 + t)) + (t_1 - (sqrt(z) + sqrt(t))));
	elseif (y <= 2.05e-35)
		tmp = 2.0 + (t_1 - (sqrt(y) + sqrt(z)));
	elseif (y <= 4.5e+15)
		tmp = ((t_2 + sqrt((1.0 + y))) - sqrt(y)) - sqrt(x);
	else
		tmp = 1.0 / (sqrt(x) + t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 4.2e-215], N[(2.0 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e-35], N[(2.0 + N[(t$95$1 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+15], N[(N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.2e-215

   1. Initial program 95.1%

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

      1. associate-+l+ 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. associate-+l- 61.4%

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

      6. associate-+r- 57.9%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. associate--l+ 51.0%

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

      2. associate--l+ 52.6%

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

      3. associate-+r+ 52.6%

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

      4. +-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in y around 0 30.4%

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

      1. +-commutative 30.4%

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

      2. associate--l+ 64.9%

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

      3. associate--l+ 61.2%

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

      4. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.2e-215 < y < 2.05000000000000013e-35

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. +-commutative 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. associate-+l- 54.6%

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

      6. associate-+r- 51.1%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in t around inf 21.2%

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

   5. Taylor expanded in y around 0 21.2%

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

   6. Taylor expanded in x around 0 27.5%

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

      1. associate--l+ 54.6%

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

      2. +-commutative 54.6%

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

   8. Simplified 54.6%

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

   if 2.05000000000000013e-35 < y < 4.5e15

   1. Initial program 92.6%

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

      1. associate-+l+ 92.6%

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

      2. +-commutative 92.6%

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

      3. associate-+l+ 92.6%

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

      4. +-commutative 92.6%

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

      5. associate-+l- 50.6%

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

      6. associate-+r- 47.3%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in t around inf 12.5%

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

   5. Taylor expanded in z around inf 18.5%

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

      1. +-commutative 18.5%

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

   7. Simplified 18.5%

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

      1. associate-+r- 18.5%

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

      2. associate--r+ 18.5%

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

      3. +-commutative 18.5%

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

      4. +-commutative 18.5%

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

   9. Applied egg-rr 18.5%

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

   if 4.5e15 < y

   1. Initial program 86.2%

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

      1. associate-+l+ 86.2%

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

      2. +-commutative 86.2%

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

      3. associate-+l+ 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-+l- 66.4%

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

      6. associate-+r- 56.3%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in t around inf 20.3%

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

   5. Taylor expanded in z around inf 19.1%

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

      1. +-commutative 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in y around inf 18.5%

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

      1. flip-- 18.5%

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

      2. div-inv 18.5%

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

      3. +-commutative 18.5%

         \[\leadsto \left(\sqrt{\color{blue}{x + 1}} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      4. +-commutative 18.5%

         \[\leadsto \left(\sqrt{x + 1} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      5. rem-square-sqrt 18.9%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      6. +-commutative 18.9%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      7. rem-square-sqrt 18.5%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

   10. Applied egg-rr 18.5%

       \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.5%

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. *-rgt-identity 18.5%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. associate--l+ 23.8%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-inverses 23.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       5. metadata-eval 23.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

   12. Simplified 23.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{t}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 14: 38.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 3.5e-23)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z))))
     (if (<= y 4.5e+15)
       (+ t_1 (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 3.5e-23) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	} else if (y <= 4.5e+15) {
		tmp = t_1 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 3.5d-23) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z)))
    else if (y <= 4.5d+15) then
        tmp = t_1 + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 3.5e-23) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (y <= 4.5e+15) {
		tmp = t_1 + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 3.5e-23:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z)))
	elif y <= 4.5e+15:
		tmp = t_1 + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 3.5e-23)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z))));
	elseif (y <= 4.5e+15)
		tmp = Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 3.5e-23)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	elseif (y <= 4.5e+15)
		tmp = t_1 + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.5e-23], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+15], 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[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.49999999999999993e-23

   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. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 56.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 53.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 21.4%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in x around 0 24.5%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 51.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative 51.9%

         \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   8. Simplified 51.9%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 3.49999999999999993e-23 < y < 4.5e15

   1. Initial program 91.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 91.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 91.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 91.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 91.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 50.5%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 46.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 9.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 21.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 21.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 21.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 4.5e15 < y

   1. Initial program 86.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.2%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 66.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 56.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 20.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 19.1%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 18.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   9. Step-by-step derivation

      1. flip-- 18.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

      2. div-inv 18.5%

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

      3. +-commutative 18.5%

         \[\leadsto \left(\sqrt{\color{blue}{x + 1}} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      4. +-commutative 18.5%

         \[\leadsto \left(\sqrt{x + 1} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      5. rem-square-sqrt 18.9%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      6. +-commutative 18.9%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      7. rem-square-sqrt 18.5%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

   10. Applied egg-rr 18.5%

       \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.5%

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. *-rgt-identity 18.5%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. associate--l+ 23.8%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-inverses 23.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       5. metadata-eval 23.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

   12. Simplified 23.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 15: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 3.2 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 3.2e-23)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z))))
     (if (<= y 1.6e+41)
       (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- t_1 (sqrt x)))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 3.2e-23) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	} else if (y <= 1.6e+41) {
		tmp = (sqrt((1.0 + y)) - sqrt(y)) + (t_1 - sqrt(x));
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 3.2d-23) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z)))
    else if (y <= 1.6d+41) then
        tmp = (sqrt((1.0d0 + y)) - sqrt(y)) + (t_1 - sqrt(x))
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 3.2e-23) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (y <= 1.6e+41) {
		tmp = (Math.sqrt((1.0 + y)) - Math.sqrt(y)) + (t_1 - Math.sqrt(x));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 3.2e-23:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z)))
	elif y <= 1.6e+41:
		tmp = (math.sqrt((1.0 + y)) - math.sqrt(y)) + (t_1 - math.sqrt(x))
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 3.2e-23)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z))));
	elseif (y <= 1.6e+41)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(t_1 - sqrt(x)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 3.2e-23)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	elseif (y <= 1.6e+41)
		tmp = (sqrt((1.0 + y)) - sqrt(y)) + (t_1 - sqrt(x));
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 3.2e-23], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+41], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.19999999999999976e-23

   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. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 56.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 53.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 21.4%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in x around 0 24.5%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 51.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative 51.9%

         \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   8. Simplified 51.9%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 3.19999999999999976e-23 < y < 1.60000000000000005e41

   1. Initial program 89.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 89.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 89.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 89.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 89.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 58.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 53.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 24.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 20.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 20.9%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 20.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 20.9%

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{x + 1}} \]

      2. associate--r+ 20.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)} + \sqrt{x + 1} \]

      3. associate-+l- 32.7%

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{x + 1}\right)} \]

      4. +-commutative 32.7%

         \[\leadsto \left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{\color{blue}{1 + x}}\right) \]

   9. Applied egg-rr 32.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} - \sqrt{1 + x}\right)} \]

   if 1.60000000000000005e41 < y

   1. Initial program 86.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 66.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 55.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 18.9%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 18.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 18.8%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 18.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   9. Step-by-step derivation

      1. flip-- 18.2%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

      2. div-inv 18.2%

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

      3. +-commutative 18.2%

         \[\leadsto \left(\sqrt{\color{blue}{x + 1}} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      4. +-commutative 18.2%

         \[\leadsto \left(\sqrt{x + 1} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      5. rem-square-sqrt 18.6%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      6. +-commutative 18.6%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      7. rem-square-sqrt 18.2%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

   10. Applied egg-rr 18.2%

       \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.2%

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. *-rgt-identity 18.2%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. associate--l+ 24.0%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-inverses 24.0%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       5. metadata-eval 24.0%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

   12. Simplified 24.0%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 16: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(t_1 + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 2.05e-35)
     (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z))))
     (if (<= y 5e+14)
       (- (- (+ t_1 (sqrt (+ 1.0 y))) (sqrt y)) (sqrt x))
       (/ 1.0 (+ (sqrt x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 2.05e-35) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	} else if (y <= 5e+14) {
		tmp = ((t_1 + sqrt((1.0 + y))) - sqrt(y)) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt(x) + t_1);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    if (y <= 2.05d-35) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z)))
    else if (y <= 5d+14) then
        tmp = ((t_1 + sqrt((1.0d0 + y))) - sqrt(y)) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt(x) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 2.05e-35) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (y <= 5e+14) {
		tmp = ((t_1 + Math.sqrt((1.0 + y))) - Math.sqrt(y)) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt(x) + t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 2.05e-35:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z)))
	elif y <= 5e+14:
		tmp = ((t_1 + math.sqrt((1.0 + y))) - math.sqrt(y)) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt(x) + t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 2.05e-35)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z))));
	elseif (y <= 5e+14)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) - sqrt(y)) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 2.05e-35)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	elseif (y <= 5e+14)
		tmp = ((t_1 + sqrt((1.0 + y))) - sqrt(y)) - sqrt(x);
	else
		tmp = 1.0 / (sqrt(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 2.05e-35], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+14], N[(N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.05000000000000013e-35

   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. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 57.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 53.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 25.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 21.3%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in x around 0 24.1%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 51.7%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative 51.7%

         \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   8. Simplified 51.7%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 2.05000000000000013e-35 < y < 5e14

   1. Initial program 92.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 92.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 92.6%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 92.6%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 92.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 50.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 47.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 9.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 12.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 18.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 18.5%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 18.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-+r- 18.5%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

      2. associate--r+ 18.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}} \]

      3. +-commutative 18.5%

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{x + 1}\right)} - \sqrt{y}\right) - \sqrt{x} \]

      4. +-commutative 18.5%

         \[\leadsto \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{y}\right) - \sqrt{x} \]

   9. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \sqrt{y}\right) - \sqrt{x}} \]

   if 5e14 < y

   1. Initial program 86.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.2%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.2%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.2%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 66.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 56.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 20.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 19.1%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 18.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   9. Step-by-step derivation

      1. flip-- 18.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

      2. div-inv 18.5%

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

      3. +-commutative 18.5%

         \[\leadsto \left(\sqrt{\color{blue}{x + 1}} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      4. +-commutative 18.5%

         \[\leadsto \left(\sqrt{x + 1} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      5. rem-square-sqrt 18.9%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      6. +-commutative 18.9%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      7. rem-square-sqrt 18.5%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

   10. Applied egg-rr 18.5%

       \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.5%

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. *-rgt-identity 18.5%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. associate--l+ 23.8%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-inverses 23.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       5. metadata-eval 23.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

   12. Simplified 23.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 17: 40.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 42000000000:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.45e-23)
   (+ 2.0 (- (sqrt (+ 1.0 z)) (+ (sqrt y) (sqrt z))))
   (if (<= y 42000000000.0)
     (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
     (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.45e-23) {
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	} else if (y <= 42000000000.0) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= 2.45d-23) then
        tmp = 2.0d0 + (sqrt((1.0d0 + z)) - (sqrt(y) + sqrt(z)))
    else if (y <= 42000000000.0d0) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.45e-23) {
		tmp = 2.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(y) + Math.sqrt(z)));
	} else if (y <= 42000000000.0) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.45e-23:
		tmp = 2.0 + (math.sqrt((1.0 + z)) - (math.sqrt(y) + math.sqrt(z)))
	elif y <= 42000000000.0:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.45e-23)
		tmp = Float64(2.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(y) + sqrt(z))));
	elseif (y <= 42000000000.0)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.45e-23)
		tmp = 2.0 + (sqrt((1.0 + z)) - (sqrt(y) + sqrt(z)));
	elseif (y <= 42000000000.0)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt(x) + sqrt((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.45e-23], N[(2.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 42000000000.0], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.4499999999999999e-23

   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. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 56.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 53.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.4%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 21.4%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in x around 0 24.5%

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 51.9%

         \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative 51.9%

         \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right) \]

   8. Simplified 51.9%

      \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \sqrt{y}\right)\right)} \]

   if 2.4499999999999999e-23 < y < 4.2e10

   1. Initial program 92.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 92.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 92.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 92.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 92.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 49.0%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 44.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 7.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 9.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 21.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 21.3%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 21.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in x around 0 49.2%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   9. Step-by-step derivation

      1. associate--l+ 49.2%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   if 4.2e10 < y

   1. Initial program 86.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 66.5%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 56.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 27.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 20.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 19.1%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 18.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   9. Step-by-step derivation

      1. flip-- 18.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

      2. div-inv 18.5%

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

      3. +-commutative 18.5%

         \[\leadsto \left(\sqrt{\color{blue}{x + 1}} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      4. +-commutative 18.5%

         \[\leadsto \left(\sqrt{x + 1} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      5. rem-square-sqrt 18.9%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      6. +-commutative 18.9%

         \[\leadsto \left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

      7. rem-square-sqrt 18.5%

         \[\leadsto \left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]

   10. Applied egg-rr 18.5%

       \[\leadsto \color{blue}{\left(\left(1 + x\right) - x\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.5%

          \[\leadsto \color{blue}{\frac{\left(\left(1 + x\right) - x\right) \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. *-rgt-identity 18.5%

          \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. associate--l+ 23.7%

          \[\leadsto \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-inverses 23.7%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       5. metadata-eval 23.7%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

   12. Simplified 23.7%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-23}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 42000000000:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternative 18: 28.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;y \leq 6.8:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 x)) (sqrt x)))) (if (<= y 6.8) (+ 1.0 t_1) t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (y <= 6.8) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x)) - sqrt(x)
    if (y <= 6.8d0) then
        tmp = 1.0d0 + t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (y <= 6.8) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if y <= 6.8:
		tmp = 1.0 + t_1
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (y <= 6.8)
		tmp = Float64(1.0 + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (y <= 6.8)
		tmp = 1.0 + t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 6.79999999999999982

   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. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 97.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 57.4%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 23.6%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 20.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 21.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 21.2%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 21.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around 0 20.8%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + x}\right) - \sqrt{x}} \]
   9. Step-by-step derivation

      1. associate--l+ 36.1%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   10. Simplified 36.1%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + x} - \sqrt{x}\right)} \]

   if 6.79999999999999982 < y

   1. Initial program 86.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 64.7%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.7%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 26.5%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 20.0%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 19.7%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 18.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8:\\ \;\;\;\;1 + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 19: 40.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.35:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.35)
   (- 3.0 (+ (sqrt y) (sqrt z)))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.35) {
		tmp = 3.0 - (sqrt(y) + sqrt(z));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= 0.35d0) then
        tmp = 3.0d0 - (sqrt(y) + sqrt(z))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.35) {
		tmp = 3.0 - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 0.35:
		tmp = 3.0 - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.35)
		tmp = Float64(3.0 - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.35)
		tmp = 3.0 - (sqrt(y) + sqrt(z));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 0.35], N[(3.0 - N[(N[Sqrt[y], $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]]

Derivation

1. Split input into 2 regimes

2. if z < 0.34999999999999998

   1. Initial program 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 96.9%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 96.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 96.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 57.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.4%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 14.4%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in z around 0 14.4%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{2} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. Taylor expanded in x around 0 21.0%

      \[\leadsto \color{blue}{3 - \left(\sqrt{y} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 21.0%

         \[\leadsto 3 - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} \]

   9. Simplified 21.0%

      \[\leadsto \color{blue}{3 - \left(\sqrt{z} + \sqrt{y}\right)} \]

   if 0.34999999999999998 < z

   1. Initial program 86.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.1%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.1%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 64.3%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.5%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 16.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 28.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 28.5%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 28.5%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in x around 0 33.6%

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right) - \sqrt{y}} \]
   9. Step-by-step derivation

      1. associate--l+ 54.7%

         \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]

   10. Simplified 54.7%

       \[\leadsto \color{blue}{1 + \left(\sqrt{1 + y} - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.35:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 20: 24.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.5:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.5) (- 3.0 (+ (sqrt y) (sqrt z))) (- (+ 1.0 (* x 0.5)) (sqrt x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.5) {
		tmp = 3.0 - (sqrt(y) + sqrt(z));
	} else {
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= 1.5d0) then
        tmp = 3.0d0 - (sqrt(y) + sqrt(z))
    else
        tmp = (1.0d0 + (x * 0.5d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.5) {
		tmp = 3.0 - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = (1.0 + (x * 0.5)) - Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.5:
		tmp = 3.0 - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = (1.0 + (x * 0.5)) - math.sqrt(x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.5)
		tmp = Float64(3.0 - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.5)
		tmp = 3.0 - (sqrt(y) + sqrt(z));
	else
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.5], N[(3.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5

   1. Initial program 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 96.9%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 96.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 96.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 57.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 34.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 14.3%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in z around 0 14.3%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{2} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. Taylor expanded in x around 0 20.9%

      \[\leadsto \color{blue}{3 - \left(\sqrt{y} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 20.9%

         \[\leadsto 3 - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} \]

   9. Simplified 20.9%

      \[\leadsto \color{blue}{3 - \left(\sqrt{z} + \sqrt{y}\right)} \]

   if 1.5 < z

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.0%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.0%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 64.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 16.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 28.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 28.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 17.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   9. Taylor expanded in x around 0 18.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right)} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]

Alternative 21: 24.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.5:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.5) (- 3.0 (+ (sqrt y) (sqrt z))) (- (sqrt (+ 1.0 x)) (sqrt x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.5) {
		tmp = 3.0 - (sqrt(y) + sqrt(z));
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= 1.5d0) then
        tmp = 3.0d0 - (sqrt(y) + sqrt(z))
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.5) {
		tmp = 3.0 - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.5:
		tmp = 3.0 - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.5)
		tmp = Float64(3.0 - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.5)
		tmp = 3.0 - (sqrt(y) + sqrt(z));
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.5], N[(3.0 - N[(N[Sqrt[y], $MachinePrecision] + 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 < 1.5

   1. Initial program 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 96.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 96.9%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 96.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 96.9%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 57.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 34.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 21.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in y around 0 14.3%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. Taylor expanded in z around 0 14.3%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{2} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. Taylor expanded in x around 0 20.9%

      \[\leadsto \color{blue}{3 - \left(\sqrt{y} + \sqrt{z}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 20.9%

         \[\leadsto 3 - \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} \]

   9. Simplified 20.9%

      \[\leadsto \color{blue}{3 - \left(\sqrt{z} + \sqrt{y}\right)} \]

   if 1.5 < z

   1. Initial program 86.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 86.0%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative 86.0%

         \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. associate-+l+ 86.0%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      4. +-commutative 86.0%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

      5. associate-+l- 64.6%

         \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

      6. associate-+r- 54.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   3. Simplified 16.1%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

   4. Taylor expanded in t around inf 19.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   5. Taylor expanded in z around inf 28.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 28.6%

         \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   7. Simplified 28.6%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in y around inf 17.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5:\\ \;\;\;\;3 - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 22: 16.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot 0.5\right) - \sqrt{x} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (+ 1.0 (* x 0.5)) (sqrt x)))

C

double code(double x, double y, double z, double t) {
	return (1.0 + (x * 0.5)) - sqrt(x);
}

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 + (x * 0.5d0)) - sqrt(x)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 + (x * 0.5)) - Math.sqrt(x);
}

Python

def code(x, y, z, t):
	return (1.0 + (x * 0.5)) - math.sqrt(x)

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 + (x * 0.5)) - sqrt(x);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 91.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. +-commutative 91.4%

      \[\leadsto \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. associate-+l+ 91.4%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   4. +-commutative 91.4%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

   5. associate-+l- 61.1%

      \[\leadsto \left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{t + 1} - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)\right)} \]

   6. associate-+r- 54.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \sqrt{t + 1}\right) - \left(\sqrt{t} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} \]

3. Simplified 25.1%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + t} - \sqrt{x}\right) - \left(\left(\sqrt{t} - \sqrt{1 + z}\right) + \left(\sqrt{y} + \left(\sqrt{z} - \sqrt{1 + y}\right)\right)\right)\right)} \]

4. Taylor expanded in t around inf 20.1%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

5. Taylor expanded in z around inf 20.4%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 20.4%

      \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

7. Simplified 20.4%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

8. Taylor expanded in y around inf 14.4%

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

9. Taylor expanded in x around 0 15.1%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right)} - \sqrt{x} \]

10. Final simplification 15.1%

    \[\leadsto \left(1 + x \cdot 0.5\right) - \sqrt{x} \]

Developer target: 98.0% 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 2023297 
(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))))