Main:z from

Percentage Accurate: 91.4% → 99.7%

Time: 33.1s

Alternatives: 17

Speedup: 2.0×

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: 91.4% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;t_1 - \sqrt{z} \leq 0.97:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{t_2 + \sqrt{y}} + \frac{1}{\sqrt{z} + t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(t_2 + t_1\right)\right) - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= (- t_1 (sqrt z)) 0.97)
     (+
      (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
      (+ (/ 1.0 (+ t_2 (sqrt y))) (/ 1.0 (+ (sqrt z) t_1))))
     (+
      (- (+ 1.0 (+ t_2 t_1)) (sqrt z))
      (- (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if ((t_1 - sqrt(z)) <= 0.97) {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt(z) + t_1)));
	} else {
		tmp = ((1.0 + (t_2 + t_1)) - sqrt(z)) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + y))
    if ((t_1 - sqrt(z)) <= 0.97d0) then
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + ((1.0d0 / (t_2 + sqrt(y))) + (1.0d0 / (sqrt(z) + t_1)))
    else
        tmp = ((1.0d0 + (t_2 + t_1)) - sqrt(z)) + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if ((t_1 - Math.sqrt(z)) <= 0.97) {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + ((1.0 / (t_2 + Math.sqrt(y))) + (1.0 / (Math.sqrt(z) + t_1)));
	} else {
		tmp = ((1.0 + (t_2 + t_1)) - Math.sqrt(z)) + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if (t_1 - math.sqrt(z)) <= 0.97:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + ((1.0 / (t_2 + math.sqrt(y))) + (1.0 / (math.sqrt(z) + t_1)))
	else:
		tmp = ((1.0 + (t_2 + t_1)) - math.sqrt(z)) + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 0.97)
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(1.0 / Float64(sqrt(z) + t_1))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(t_2 + t_1)) - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if ((t_1 - sqrt(z)) <= 0.97)
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 / (t_2 + sqrt(y))) + (1.0 / (sqrt(z) + t_1)));
	else
		tmp = ((1.0 + (t_2 + t_1)) - sqrt(z)) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.97], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z)) < 0.96999999999999997

   1. Initial program 89.2%

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

      1. associate-+l+ 89.2%

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

      2. associate-+l+ 89.2%

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

      3. +-commutative 89.2%

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

      4. +-commutative 89.2%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in t around inf 41.5%

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

      1. flip-- 42.1%

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

      2. add-sqr-sqrt 33.9%

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

      3. +-commutative 33.9%

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

      4. add-sqr-sqrt 42.1%

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

      5. +-commutative 42.1%

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

   6. Applied egg-rr 42.1%

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

      1. associate--l+ 43.6%

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

      2. +-inverses 43.6%

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

      3. metadata-eval 43.6%

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

   8. Simplified 43.6%

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

      1. flip-- 43.7%

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

      2. add-sqr-sqrt 38.7%

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

      3. add-sqr-sqrt 43.7%

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

   10. Applied egg-rr 43.7%

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

       1. associate--l+ 46.3%

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

       2. +-inverses 46.3%

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

       3. metadata-eval 46.3%

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

   12. Simplified 46.3%

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

       1. flip-- 46.6%

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

       2. add-sqr-sqrt 26.8%

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

       3. add-sqr-sqrt 46.8%

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

   14. Applied egg-rr 46.8%

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

       1. associate--l+ 50.0%

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

       2. +-inverses 50.0%

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

       3. metadata-eval 50.0%

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

   16. Simplified 50.0%

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

   if 0.96999999999999997 < (-.f64 (sqrt.f64 (+.f64 z 1)) (sqrt.f64 z))

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-+r+ 97.9%

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

      3. associate-+r- 75.6%

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

      4. associate-+l- 65.2%

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

      5. associate-+r- 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in x around 0 33.7%

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

      1. flip-- 34.0%

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

      2. add-sqr-sqrt 25.5%

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

      3. add-sqr-sqrt 34.0%

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

   6. Applied egg-rr 34.0%

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

      1. associate--l+ 34.0%

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

      2. +-inverses 34.0%

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

      3. metadata-eval 34.0%

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

   8. Simplified 34.0%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.97:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{y}\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ \mathbf{if}\;t \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(1 + \left(1 + t_1\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))))
   (if (<= t 2.1e+20)
     (+
      (- (+ 1.0 (+ 1.0 t_1)) (sqrt z))
      (- (- (sqrt (+ 1.0 t)) (sqrt t)) (sqrt y)))
     (+
      (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
      (+ (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (- t_1 (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double tmp;
	if (t <= 2.1e+20) {
		tmp = ((1.0 + (1.0 + t_1)) - sqrt(z)) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    if (t <= 2.1d+20) then
        tmp = ((1.0d0 + (1.0d0 + t_1)) - sqrt(z)) + ((sqrt((1.0d0 + t)) - sqrt(t)) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) + (t_1 - sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double tmp;
	if (t <= 2.1e+20) {
		tmp = ((1.0 + (1.0 + t_1)) - Math.sqrt(z)) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) + (t_1 - Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	tmp = 0
	if t <= 2.1e+20:
		tmp = ((1.0 + (1.0 + t_1)) - math.sqrt(z)) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) + (t_1 - math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (t <= 2.1e+20)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 + t_1)) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) + Float64(t_1 - sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	tmp = 0.0;
	if (t <= 2.1e+20)
		tmp = ((1.0 + (1.0 + t_1)) - sqrt(z)) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	else
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) + (t_1 - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2.1e+20], N[(N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e20

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-+r- 74.0%

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

      4. associate-+l- 54.2%

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

      5. associate-+r- 49.9%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in x around 0 24.1%

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

   5. Taylor expanded in y around 0 21.2%

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

   if 2.1e20 < t

   1. Initial program 89.7%

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

      1. associate-+l+ 89.7%

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

      2. associate-+l+ 89.7%

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

      3. +-commutative 89.7%

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

      4. +-commutative 89.7%

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

      5. associate-+l- 43.1%

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

      6. +-commutative 43.1%

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

      7. +-commutative 43.1%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in t around inf 89.7%

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

      1. flip-- 90.4%

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

      2. add-sqr-sqrt 70.4%

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

      3. +-commutative 70.4%

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

      4. add-sqr-sqrt 90.4%

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

      5. +-commutative 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. associate--l+ 91.8%

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

      2. +-inverses 91.8%

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

      3. metadata-eval 91.8%

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

   8. Simplified 91.8%

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

      1. flip-- 91.8%

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

      2. add-sqr-sqrt 74.0%

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

      3. add-sqr-sqrt 91.8%

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

   10. Applied egg-rr 91.8%

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

       1. associate--l+ 95.1%

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

       2. +-inverses 95.1%

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

       3. metadata-eval 95.1%

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

   12. Simplified 95.1%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(1 + \left(1 + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;t \leq 1.3 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(1 + \left(t_2 + t_1\right)\right) - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{t_2 + \sqrt{y}} + \left(t_1 - \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= t 1.3e+39)
     (+
      (- (+ 1.0 (+ t_2 t_1)) (sqrt z))
      (- (/ 1.0 (+ (sqrt (+ 1.0 t)) (sqrt t))) (sqrt y)))
     (+
      (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
      (+ (/ 1.0 (+ t_2 (sqrt y))) (- t_1 (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (t <= 1.3e+39) {
		tmp = ((1.0 + (t_2 + t_1)) - sqrt(z)) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 / (t_2 + sqrt(y))) + (t_1 - sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((1.0d0 + y))
    if (t <= 1.3d+39) then
        tmp = ((1.0d0 + (t_2 + t_1)) - sqrt(z)) + ((1.0d0 / (sqrt((1.0d0 + t)) + sqrt(t))) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + ((1.0d0 / (t_2 + sqrt(y))) + (t_1 - sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (t <= 1.3e+39) {
		tmp = ((1.0 + (t_2 + t_1)) - Math.sqrt(z)) + ((1.0 / (Math.sqrt((1.0 + t)) + Math.sqrt(t))) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + ((1.0 / (t_2 + Math.sqrt(y))) + (t_1 - Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if t <= 1.3e+39:
		tmp = ((1.0 + (t_2 + t_1)) - math.sqrt(z)) + ((1.0 / (math.sqrt((1.0 + t)) + math.sqrt(t))) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + ((1.0 / (t_2 + math.sqrt(y))) + (t_1 - math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t <= 1.3e+39)
		tmp = Float64(Float64(Float64(1.0 + Float64(t_2 + t_1)) - sqrt(z)) + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + t)) + sqrt(t))) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(Float64(1.0 / Float64(t_2 + sqrt(y))) + Float64(t_1 - sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (t <= 1.3e+39)
		tmp = ((1.0 + (t_2 + t_1)) - sqrt(z)) + ((1.0 / (sqrt((1.0 + t)) + sqrt(t))) - sqrt(y));
	else
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 / (t_2 + sqrt(y))) + (t_1 - sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.3e+39], N[(N[(N[(1.0 + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e39

   1. Initial program 96.8%

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

      1. +-commutative 96.8%

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

      2. associate-+r+ 96.8%

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

      3. associate-+r- 73.7%

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

      4. associate-+l- 55.2%

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

      5. associate-+r- 50.3%

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

   3. Simplified 32.0%

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

   4. Taylor expanded in x around 0 24.4%

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

      1. flip-- 24.7%

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

      2. add-sqr-sqrt 23.4%

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

      3. add-sqr-sqrt 24.7%

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

   6. Applied egg-rr 24.7%

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

      1. associate--l+ 24.9%

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

      2. +-inverses 24.9%

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

      3. metadata-eval 24.9%

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

   8. Simplified 24.9%

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

   if 1.3e39 < t

   1. Initial program 88.8%

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

      1. associate-+l+ 88.8%

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

      2. associate-+l+ 88.8%

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

      3. +-commutative 88.8%

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

      4. +-commutative 88.8%

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

      5. associate-+l- 41.2%

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

      6. +-commutative 41.2%

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

      7. +-commutative 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in t around inf 88.8%

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

      1. flip-- 89.5%

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

      2. add-sqr-sqrt 69.4%

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

      3. +-commutative 69.4%

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

      4. add-sqr-sqrt 89.6%

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

      5. +-commutative 89.6%

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

   6. Applied egg-rr 89.6%

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

      1. associate--l+ 91.2%

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

      2. +-inverses 91.2%

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

      3. metadata-eval 91.2%

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

   8. Simplified 91.2%

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

      1. flip-- 91.2%

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

      2. add-sqr-sqrt 73.3%

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

      3. add-sqr-sqrt 91.2%

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

   10. Applied egg-rr 91.2%

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

       1. associate--l+ 94.8%

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

       2. +-inverses 94.8%

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

       3. metadata-eval 94.8%

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

   12. Simplified 94.8%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 3.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(1 + \left(t_1 + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{t_1 + \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 3.4e+19)
     (+
      (- (+ 1.0 (+ t_1 (sqrt (+ z 1.0)))) (sqrt z))
      (- (- (sqrt (+ 1.0 t)) (sqrt t)) (sqrt y)))
     (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (/ 1.0 (+ t_1 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (z <= 3.4e+19) {
		tmp = ((1.0 + (t_1 + sqrt((z + 1.0)))) - sqrt(z)) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (1.0 / (t_1 + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 3.4d+19) then
        tmp = ((1.0d0 + (t_1 + sqrt((z + 1.0d0)))) - sqrt(z)) + ((sqrt((1.0d0 + t)) - sqrt(t)) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (1.0d0 / (t_1 + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 3.4e+19) {
		tmp = ((1.0 + (t_1 + Math.sqrt((z + 1.0)))) - Math.sqrt(z)) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (1.0 / (t_1 + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 3.4e+19:
		tmp = ((1.0 + (t_1 + math.sqrt((z + 1.0)))) - math.sqrt(z)) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (1.0 / (t_1 + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 3.4e+19)
		tmp = Float64(Float64(Float64(1.0 + Float64(t_1 + sqrt(Float64(z + 1.0)))) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(1.0 / Float64(t_1 + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 3.4e+19)
		tmp = ((1.0 + (t_1 + sqrt((z + 1.0)))) - sqrt(z)) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	else
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (1.0 / (t_1 + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 3.4e+19], N[(N[(N[(1.0 + N[(t$95$1 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4e19

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 74.8%

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

      4. associate-+l- 64.5%

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

      5. associate-+r- 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in x around 0 33.9%

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

   if 3.4e19 < z

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. associate-+l+ 89.4%

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

      3. +-commutative 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-+l- 89.4%

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

      6. +-commutative 89.4%

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

      7. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in t around inf 41.0%

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

      1. flip-- 41.6%

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

      2. add-sqr-sqrt 33.8%

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

      3. +-commutative 33.8%

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

      4. add-sqr-sqrt 41.7%

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

      5. +-commutative 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. associate--l+ 43.3%

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

      2. +-inverses 43.3%

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

      3. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. flip-- 43.3%

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

      2. add-sqr-sqrt 38.9%

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

      3. add-sqr-sqrt 43.3%

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

   10. Applied egg-rr 43.3%

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

       1. associate--l+ 46.2%

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

       2. +-inverses 46.2%

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

       3. metadata-eval 46.2%

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

   12. Simplified 46.2%

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

   13. Taylor expanded in z around inf 46.2%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(1 + \left(1 + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.4e+19)
   (+
    (- (+ 1.0 (+ 1.0 (sqrt (+ z 1.0)))) (sqrt z))
    (- (- (sqrt (+ 1.0 t)) (sqrt t)) (sqrt y)))
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.4e+19) {
		tmp = ((1.0 + (1.0 + sqrt((z + 1.0)))) - sqrt(z)) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.4d+19) then
        tmp = ((1.0d0 + (1.0d0 + sqrt((z + 1.0d0)))) - sqrt(z)) + ((sqrt((1.0d0 + t)) - sqrt(t)) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.4e+19) {
		tmp = ((1.0 + (1.0 + Math.sqrt((z + 1.0)))) - Math.sqrt(z)) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3.4e+19:
		tmp = ((1.0 + (1.0 + math.sqrt((z + 1.0)))) - math.sqrt(z)) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.4e+19)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 + sqrt(Float64(z + 1.0)))) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.4e+19)
		tmp = ((1.0 + (1.0 + sqrt((z + 1.0)))) - sqrt(z)) + ((sqrt((1.0 + t)) - sqrt(t)) - sqrt(y));
	else
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3.4e+19], N[(N[(N[(1.0 + N[(1.0 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4e19

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. associate-+r+ 97.1%

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

      3. associate-+r- 74.8%

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

      4. associate-+l- 64.5%

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

      5. associate-+r- 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in x around 0 33.9%

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

   5. Taylor expanded in y around 0 28.8%

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

   if 3.4e19 < z

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. associate-+l+ 89.4%

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

      3. +-commutative 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-+l- 89.4%

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

      6. +-commutative 89.4%

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

      7. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in t around inf 41.0%

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

      1. flip-- 41.6%

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

      2. add-sqr-sqrt 33.8%

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

      3. +-commutative 33.8%

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

      4. add-sqr-sqrt 41.7%

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

      5. +-commutative 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. associate--l+ 43.3%

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

      2. +-inverses 43.3%

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

      3. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. flip-- 43.3%

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

      2. add-sqr-sqrt 38.9%

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

      3. add-sqr-sqrt 43.3%

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

   10. Applied egg-rr 43.3%

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

       1. associate--l+ 46.2%

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

       2. +-inverses 46.2%

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

       3. metadata-eval 46.2%

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

   12. Simplified 46.2%

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

   13. Taylor expanded in z around inf 46.2%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(1 + \left(1 + \sqrt{z + 1}\right)\right) - \sqrt{z}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \]

Alternative 6: 92.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 1.6 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + t_1\right) + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{t_1 + \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))))
   (if (<= z 1.6e+19)
     (+ (+ 1.0 t_1) (- (sqrt (+ z 1.0)) (+ (sqrt z) (sqrt y))))
     (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) (/ 1.0 (+ t_1 (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (z <= 1.6e+19) {
		tmp = (1.0 + t_1) + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (1.0 / (t_1 + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    if (z <= 1.6d+19) then
        tmp = (1.0d0 + t_1) + (sqrt((z + 1.0d0)) - (sqrt(z) + sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + (1.0d0 / (t_1 + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double tmp;
	if (z <= 1.6e+19) {
		tmp = (1.0 + t_1) + (Math.sqrt((z + 1.0)) - (Math.sqrt(z) + Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + (1.0 / (t_1 + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 1.6e+19:
		tmp = (1.0 + t_1) + (math.sqrt((z + 1.0)) - (math.sqrt(z) + math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + (1.0 / (t_1 + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (z <= 1.6e+19)
		tmp = Float64(Float64(1.0 + t_1) + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(1.0 / Float64(t_1 + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	tmp = 0.0;
	if (z <= 1.6e+19)
		tmp = (1.0 + t_1) + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(y)));
	else
		tmp = (1.0 / (sqrt((1.0 + x)) + sqrt(x))) + (1.0 / (t_1 + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 1.6e+19], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e19

   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%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in t around inf 54.7%

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

   5. Taylor expanded in x around 0 27.2%

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

      1. expm1-log1p-u 26.5%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      2. expm1-udef 26.5%

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

      3. log1p-udef 26.5%

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

      4. add-exp-log 27.2%

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

   7. Applied egg-rr 27.2%

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

      1. add-exp-log 26.5%

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

      2. log1p-udef 26.5%

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

      3. expm1-udef 26.5%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      4. expm1-log1p-u 27.2%

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

      5. associate-+r- 43.9%

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

      6. associate--l+ 43.9%

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

   9. Applied egg-rr 43.9%

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

       1. associate-+r+ 27.2%

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

   11. Simplified 27.2%

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

   if 1.6e19 < z

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. associate-+l+ 89.4%

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

      3. +-commutative 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-+l- 89.4%

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

      6. +-commutative 89.4%

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

      7. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in t around inf 41.0%

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

      1. flip-- 41.6%

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

      2. add-sqr-sqrt 33.8%

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

      3. +-commutative 33.8%

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

      4. add-sqr-sqrt 41.7%

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

      5. +-commutative 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. associate--l+ 43.3%

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

      2. +-inverses 43.3%

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

      3. metadata-eval 43.3%

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

   8. Simplified 43.3%

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

      1. flip-- 43.3%

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

      2. add-sqr-sqrt 38.9%

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

      3. add-sqr-sqrt 43.3%

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

   10. Applied egg-rr 43.3%

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

       1. associate--l+ 46.2%

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

       2. +-inverses 46.2%

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

       3. metadata-eval 46.2%

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

   12. Simplified 46.2%

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

   13. Taylor expanded in z around inf 46.2%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \end{array} \]

Alternative 7: 89.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;1 + \left(t_2 + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;t_1 + \left(t_2 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{t_1 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 y))))
   (if (<= y 5.5e-31)
     (+ 1.0 (+ t_2 (- (sqrt (+ z 1.0)) (+ (sqrt z) (sqrt y)))))
     (if (<= y 2e+15)
       (+ t_1 (- t_2 (+ (sqrt x) (sqrt y))))
       (/ (+ 1.0 (- x x)) (+ t_1 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + y));
	double tmp;
	if (y <= 5.5e-31) {
		tmp = 1.0 + (t_2 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(y))));
	} else if (y <= 2e+15) {
		tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (t_1 + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    if (y <= 5.5d-31) then
        tmp = 1.0d0 + (t_2 + (sqrt((z + 1.0d0)) - (sqrt(z) + sqrt(y))))
    else if (y <= 2d+15) then
        tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)))
    else
        tmp = (1.0d0 + (x - x)) / (t_1 + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + y));
	double tmp;
	if (y <= 5.5e-31) {
		tmp = 1.0 + (t_2 + (Math.sqrt((z + 1.0)) - (Math.sqrt(z) + Math.sqrt(y))));
	} else if (y <= 2e+15) {
		tmp = t_1 + (t_2 - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (1.0 + (x - x)) / (t_1 + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + y))
	tmp = 0
	if y <= 5.5e-31:
		tmp = 1.0 + (t_2 + (math.sqrt((z + 1.0)) - (math.sqrt(z) + math.sqrt(y))))
	elif y <= 2e+15:
		tmp = t_1 + (t_2 - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (1.0 + (x - x)) / (t_1 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (y <= 5.5e-31)
		tmp = Float64(1.0 + Float64(t_2 + Float64(sqrt(Float64(z + 1.0)) - Float64(sqrt(z) + sqrt(y)))));
	elseif (y <= 2e+15)
		tmp = Float64(t_1 + Float64(t_2 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(t_1 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + y));
	tmp = 0.0;
	if (y <= 5.5e-31)
		tmp = 1.0 + (t_2 + (sqrt((z + 1.0)) - (sqrt(z) + sqrt(y))));
	elseif (y <= 2e+15)
		tmp = t_1 + (t_2 - (sqrt(x) + sqrt(y)));
	else
		tmp = (1.0 + (x - x)) / (t_1 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 5.5e-31], N[(1.0 + N[(t$95$2 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+15], N[(t$95$1 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.49999999999999958e-31

   1. Initial program 97.7%

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

      1. associate-+l+ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l- 82.9%

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

      6. +-commutative 82.9%

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

      7. +-commutative 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in t around inf 52.6%

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

   5. Taylor expanded in x around 0 30.2%

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

      1. associate--l+ 46.0%

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

      2. associate--l+ 54.2%

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

   7. Simplified 54.2%

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

   if 5.49999999999999958e-31 < y < 2e15

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

      2. associate-+r+ 97.8%

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

      3. +-commutative 97.8%

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

      4. associate-+r+ 97.8%

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

      5. associate-+r+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l- 81.8%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in t around inf 25.9%

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

      1. associate--l+ 29.4%

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

      2. +-commutative 29.4%

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

      3. +-commutative 29.4%

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

   6. Simplified 29.4%

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

   7. Taylor expanded in z around inf 20.8%

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

      1. +-commutative 20.8%

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

   9. Simplified 20.8%

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

   if 2e15 < y

   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. +-commutative 90.4%

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

      2. associate-+r+ 90.4%

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

      3. +-commutative 90.4%

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

      4. associate-+r+ 90.4%

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

      5. associate-+r+ 90.4%

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

      6. +-commutative 90.4%

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

      7. associate-+l- 71.4%

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

   3. Simplified 11.9%

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

   4. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

   6. Simplified 19.9%

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

   7. Taylor expanded in z around inf 18.5%

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

      1. +-commutative 18.5%

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

   9. Simplified 18.5%

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

   10. Taylor expanded in y around inf 18.1%

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

       1. +-commutative 18.1%

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

       2. flip-- 18.6%

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

       3. add-sqr-sqrt 18.8%

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

       4. add-sqr-sqrt 18.6%

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

   12. Applied egg-rr 18.6%

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

       1. +-commutative 18.6%

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

       2. associate--l+ 20.9%

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

       3. +-commutative 20.9%

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

       4. +-commutative 20.9%

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

   14. Simplified 20.9%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} + \left(\sqrt{z + 1} - \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 8: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{z + 1} + \left(1 + y \cdot 0.5\right)\right)\right) + \frac{z - y}{\sqrt{y} - \sqrt{z}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3e+15)
   (+
    (+ 1.0 (+ (sqrt (+ z 1.0)) (+ 1.0 (* y 0.5))))
    (/ (- z y) (- (sqrt y) (sqrt z))))
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ 1.0 y)) (+ (sqrt x) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = (1.0 + (sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (sqrt(y) - sqrt(z)));
	} else {
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3d+15) then
        tmp = (1.0d0 + (sqrt((z + 1.0d0)) + (1.0d0 + (y * 0.5d0)))) + ((z - y) / (sqrt(y) - sqrt(z)))
    else
        tmp = sqrt((1.0d0 + x)) + (sqrt((1.0d0 + y)) - (sqrt(x) + sqrt(y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = (1.0 + (Math.sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (Math.sqrt(y) - Math.sqrt(z)));
	} else {
		tmp = Math.sqrt((1.0 + x)) + (Math.sqrt((1.0 + y)) - (Math.sqrt(x) + Math.sqrt(y)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3e+15:
		tmp = (1.0 + (math.sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (math.sqrt(y) - math.sqrt(z)))
	else:
		tmp = math.sqrt((1.0 + x)) + (math.sqrt((1.0 + y)) - (math.sqrt(x) + math.sqrt(y)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3e+15)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(z + 1.0)) + Float64(1.0 + Float64(y * 0.5)))) + Float64(Float64(z - y) / Float64(sqrt(y) - sqrt(z))));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(x) + sqrt(y))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3e+15)
		tmp = (1.0 + (sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (sqrt(y) - sqrt(z)));
	else
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + y)) - (sqrt(x) + sqrt(y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3e+15], N[(N[(1.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z - y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3e15

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in t around inf 55.0%

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

   5. Taylor expanded in x around 0 27.3%

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

   6. Taylor expanded in y around 0 25.9%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 25.9%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 25.9%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. flip-+ 25.9%

         \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}} \]

      2. add-sqr-sqrt 25.9%

         \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \frac{\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}} \]

      3. add-sqr-sqrt 25.9%

         \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \frac{y - \color{blue}{z}}{\sqrt{y} - \sqrt{z}} \]

   10. Applied egg-rr 25.9%

       \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \color{blue}{\frac{y - z}{\sqrt{y} - \sqrt{z}}} \]

   if 3e15 < z

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-+r+ 89.3%

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

      3. +-commutative 89.3%

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

      4. associate-+r+ 89.3%

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

      5. associate-+r+ 89.3%

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

      6. +-commutative 89.3%

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

      7. associate-+l- 48.3%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in t around inf 4.3%

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

      1. associate--l+ 20.0%

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

      2. +-commutative 20.0%

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

      3. +-commutative 20.0%

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

   6. Simplified 20.0%

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

   7. Taylor expanded in z around inf 28.8%

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

      1. +-commutative 28.8%

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

   9. Simplified 28.8%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{z + 1} + \left(1 + y \cdot 0.5\right)\right)\right) + \frac{z - y}{\sqrt{y} - \sqrt{z}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 9: 83.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{z + 1} + \left(1 + y \cdot 0.5\right)\right)\right) + \frac{z - y}{\sqrt{y} - \sqrt{z}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3e+15)
   (+
    (+ 1.0 (+ (sqrt (+ z 1.0)) (+ 1.0 (* y 0.5))))
    (/ (- z y) (- (sqrt y) (sqrt z))))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = (1.0 + (sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (sqrt(y) - sqrt(z)));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3d+15) then
        tmp = (1.0d0 + (sqrt((z + 1.0d0)) + (1.0d0 + (y * 0.5d0)))) + ((z - y) / (sqrt(y) - sqrt(z)))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = (1.0 + (Math.sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (Math.sqrt(y) - Math.sqrt(z)));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3e+15:
		tmp = (1.0 + (math.sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (math.sqrt(y) - math.sqrt(z)))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3e+15)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(z + 1.0)) + Float64(1.0 + Float64(y * 0.5)))) + Float64(Float64(z - y) / Float64(sqrt(y) - sqrt(z))));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3e+15)
		tmp = (1.0 + (sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) + ((z - y) / (sqrt(y) - sqrt(z)));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3e+15], N[(N[(1.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z - y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $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 < 3e15

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in t around inf 55.0%

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

   5. Taylor expanded in x around 0 27.3%

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

   6. Taylor expanded in y around 0 25.9%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 25.9%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 25.9%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   9. Step-by-step derivation

      1. flip-+ 25.9%

         \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \color{blue}{\frac{\sqrt{y} \cdot \sqrt{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}}} \]

      2. add-sqr-sqrt 25.9%

         \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \frac{\color{blue}{y} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{y} - \sqrt{z}} \]

      3. add-sqr-sqrt 25.9%

         \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \frac{y - \color{blue}{z}}{\sqrt{y} - \sqrt{z}} \]

   10. Applied egg-rr 25.9%

       \[\leadsto \left(1 + \left(\left(1 + y \cdot 0.5\right) + \sqrt{1 + z}\right)\right) - \color{blue}{\frac{y - z}{\sqrt{y} - \sqrt{z}}} \]

   if 3e15 < z

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-+l+ 89.3%

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

      3. +-commutative 89.3%

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

      4. +-commutative 89.3%

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

      5. associate-+l- 89.3%

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

      6. +-commutative 89.3%

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

      7. +-commutative 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in t around inf 40.7%

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

   5. Taylor expanded in x around 0 4.3%

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

      1. expm1-log1p-u 4.4%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      2. expm1-udef 4.4%

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

      3. log1p-udef 4.4%

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

      4. add-exp-log 4.3%

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

   7. Applied egg-rr 4.3%

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

   8. Taylor expanded in z around inf 30.7%

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

      1. associate--l+ 53.5%

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

   10. Simplified 53.5%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{z + 1} + \left(1 + y \cdot 0.5\right)\right)\right) + \frac{z - y}{\sqrt{y} - \sqrt{z}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 10: 83.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{z + 1} + \left(1 + y \cdot 0.5\right)\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3e+15)
   (- (+ 1.0 (+ (sqrt (+ z 1.0)) (+ 1.0 (* y 0.5)))) (+ (sqrt z) (sqrt y)))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = (1.0 + (sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) - (sqrt(z) + sqrt(y));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3d+15) then
        tmp = (1.0d0 + (sqrt((z + 1.0d0)) + (1.0d0 + (y * 0.5d0)))) - (sqrt(z) + sqrt(y))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = (1.0 + (Math.sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) - (Math.sqrt(z) + Math.sqrt(y));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3e+15:
		tmp = (1.0 + (math.sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) - (math.sqrt(z) + math.sqrt(y))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3e+15)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(z + 1.0)) + Float64(1.0 + Float64(y * 0.5)))) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3e+15)
		tmp = (1.0 + (sqrt((z + 1.0)) + (1.0 + (y * 0.5)))) - (sqrt(z) + sqrt(y));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3e+15], N[(N[(1.0 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $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 < 3e15

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in t around inf 55.0%

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

   5. Taylor expanded in x around 0 27.3%

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

   6. Taylor expanded in y around 0 25.9%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 25.9%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 25.9%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   if 3e15 < z

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-+l+ 89.3%

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

      3. +-commutative 89.3%

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

      4. +-commutative 89.3%

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

      5. associate-+l- 89.3%

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

      6. +-commutative 89.3%

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

      7. +-commutative 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in t around inf 40.7%

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

   5. Taylor expanded in x around 0 4.3%

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

      1. expm1-log1p-u 4.4%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      2. expm1-udef 4.4%

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

      3. log1p-udef 4.4%

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

      4. add-exp-log 4.3%

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

   7. Applied egg-rr 4.3%

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

   8. Taylor expanded in z around inf 30.7%

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

      1. associate--l+ 53.5%

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

   10. Simplified 53.5%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt{z + 1} + \left(1 + y \cdot 0.5\right)\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 11: 83.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \left(1 + \sqrt{z + 1}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.6e+19)
   (- (+ 1.0 (+ 1.0 (sqrt (+ z 1.0)))) (+ (sqrt z) (sqrt y)))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+19) {
		tmp = (1.0 + (1.0 + sqrt((z + 1.0)))) - (sqrt(z) + sqrt(y));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.6d+19) then
        tmp = (1.0d0 + (1.0d0 + sqrt((z + 1.0d0)))) - (sqrt(z) + sqrt(y))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.6e+19) {
		tmp = (1.0 + (1.0 + Math.sqrt((z + 1.0)))) - (Math.sqrt(z) + Math.sqrt(y));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.6e+19:
		tmp = (1.0 + (1.0 + math.sqrt((z + 1.0)))) - (math.sqrt(z) + math.sqrt(y))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.6e+19)
		tmp = Float64(Float64(1.0 + Float64(1.0 + sqrt(Float64(z + 1.0)))) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.6e+19)
		tmp = (1.0 + (1.0 + sqrt((z + 1.0)))) - (sqrt(z) + sqrt(y));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.6e+19], N[(N[(1.0 + N[(1.0 + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $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 < 1.6e19

   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%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in t around inf 54.7%

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

   5. Taylor expanded in x around 0 27.2%

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

   6. Taylor expanded in y around 0 23.2%

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

   if 1.6e19 < z

   1. Initial program 89.4%

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

      1. associate-+l+ 89.4%

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

      2. associate-+l+ 89.4%

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

      3. +-commutative 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-+l- 89.4%

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

      6. +-commutative 89.4%

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

      7. +-commutative 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in t around inf 41.0%

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

   5. Taylor expanded in x around 0 4.3%

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

      1. expm1-log1p-u 4.4%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      2. expm1-udef 4.4%

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

      3. log1p-udef 4.4%

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

      4. add-exp-log 4.3%

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

   7. Applied egg-rr 4.3%

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

   8. Taylor expanded in z around inf 30.9%

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

      1. associate--l+ 53.3%

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

   10. Simplified 53.3%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + \left(1 + \sqrt{z + 1}\right)\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 12: 65.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 2 + \left(y \cdot 0.5 - \sqrt{y}\right)\\ t_2 := \left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{if}\;y \leq 3.4 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (- (* y 0.5) (sqrt y))))
        (t_2 (- (+ (* y 0.5) 3.0) (sqrt y))))
   (if (<= y 3.4e-190)
     t_2
     (if (<= y 5.8e-145)
       t_1
       (if (<= y 5e-127)
         t_2
         (if (<= y 5.0) t_1 (- (sqrt (+ 1.0 x)) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 2.0 + ((y * 0.5) - sqrt(y));
	double t_2 = ((y * 0.5) + 3.0) - sqrt(y);
	double tmp;
	if (y <= 3.4e-190) {
		tmp = t_2;
	} else if (y <= 5.8e-145) {
		tmp = t_1;
	} else if (y <= 5e-127) {
		tmp = t_2;
	} else if (y <= 5.0) {
		tmp = t_1;
	} else {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 + ((y * 0.5d0) - sqrt(y))
    t_2 = ((y * 0.5d0) + 3.0d0) - sqrt(y)
    if (y <= 3.4d-190) then
        tmp = t_2
    else if (y <= 5.8d-145) then
        tmp = t_1
    else if (y <= 5d-127) then
        tmp = t_2
    else if (y <= 5.0d0) then
        tmp = t_1
    else
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 + ((y * 0.5) - Math.sqrt(y));
	double t_2 = ((y * 0.5) + 3.0) - Math.sqrt(y);
	double tmp;
	if (y <= 3.4e-190) {
		tmp = t_2;
	} else if (y <= 5.8e-145) {
		tmp = t_1;
	} else if (y <= 5e-127) {
		tmp = t_2;
	} else if (y <= 5.0) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 2.0 + ((y * 0.5) - math.sqrt(y))
	t_2 = ((y * 0.5) + 3.0) - math.sqrt(y)
	tmp = 0
	if y <= 3.4e-190:
		tmp = t_2
	elif y <= 5.8e-145:
		tmp = t_1
	elif y <= 5e-127:
		tmp = t_2
	elif y <= 5.0:
		tmp = t_1
	else:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(2.0 + Float64(Float64(y * 0.5) - sqrt(y)))
	t_2 = Float64(Float64(Float64(y * 0.5) + 3.0) - sqrt(y))
	tmp = 0.0
	if (y <= 3.4e-190)
		tmp = t_2;
	elseif (y <= 5.8e-145)
		tmp = t_1;
	elseif (y <= 5e-127)
		tmp = t_2;
	elseif (y <= 5.0)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 + ((y * 0.5) - sqrt(y));
	t_2 = ((y * 0.5) + 3.0) - sqrt(y);
	tmp = 0.0;
	if (y <= 3.4e-190)
		tmp = t_2;
	elseif (y <= 5.8e-145)
		tmp = t_1;
	elseif (y <= 5e-127)
		tmp = t_2;
	elseif (y <= 5.0)
		tmp = t_1;
	else
		tmp = sqrt((1.0 + x)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 0.5), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.4e-190], t$95$2, If[LessEqual[y, 5.8e-145], t$95$1, If[LessEqual[y, 5e-127], t$95$2, If[LessEqual[y, 5.0], t$95$1, N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.39999999999999981e-190 or 5.79999999999999968e-145 < y < 4.9999999999999997e-127

   1. Initial program 97.5%

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

      1. associate-+l+ 97.4%

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

      2. associate-+l+ 97.4%

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

      3. +-commutative 97.4%

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

      4. +-commutative 97.4%

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

      5. associate-+l- 86.2%

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

      6. +-commutative 86.2%

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

      7. +-commutative 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in t around inf 45.4%

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

   5. Taylor expanded in x around 0 28.7%

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

   6. Taylor expanded in y around 0 28.7%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 28.7%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 28.7%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   9. Taylor expanded in z around 0 44.0%

      \[\leadsto \color{blue}{\left(3 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 3.39999999999999981e-190 < y < 5.79999999999999968e-145 or 4.9999999999999997e-127 < y < 5

   1. Initial program 98.1%

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

      1. associate-+l+ 98.1%

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

      2. associate-+l+ 98.1%

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

      3. +-commutative 98.1%

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

      4. +-commutative 98.1%

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

      5. associate-+l- 79.1%

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

      6. +-commutative 79.1%

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

      7. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in t around inf 59.6%

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

   5. Taylor expanded in x around 0 32.3%

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

   6. Taylor expanded in y around 0 32.3%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 32.3%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   9. Taylor expanded in z around inf 42.4%

      \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \sqrt{y}} \]
   10. Step-by-step derivation

       1. associate--l+ 42.4%

          \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \sqrt{y}\right)} \]

       2. *-commutative 42.4%

          \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \sqrt{y}\right) \]

   11. Simplified 42.4%

       \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \sqrt{y}\right)} \]

   if 5 < y

   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. +-commutative 90.4%

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

      2. associate-+r+ 90.4%

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

      3. +-commutative 90.4%

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

      4. associate-+r+ 90.4%

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

      5. associate-+r+ 90.4%

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

      6. +-commutative 90.4%

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

      7. associate-+l- 71.8%

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

   3. Simplified 13.1%

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

   4. Taylor expanded in t around inf 4.9%

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

      1. associate--l+ 19.5%

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

      2. +-commutative 19.5%

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

      3. +-commutative 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in z around inf 18.2%

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

      1. +-commutative 18.2%

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

   9. Simplified 18.2%

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

   10. Taylor expanded in y around inf 17.8%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-190}:\\ \;\;\;\;\left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-145}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \sqrt{y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{elif}\;y \leq 5:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 13: 80.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.45:\\ \;\;\;\;\left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.45)
   (- (+ (* y 0.5) 3.0) (sqrt y))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.45) {
		tmp = ((y * 0.5) + 3.0) - sqrt(y);
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.45d0) then
        tmp = ((y * 0.5d0) + 3.0d0) - sqrt(y)
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.45) {
		tmp = ((y * 0.5) + 3.0) - Math.sqrt(y);
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.45:
		tmp = ((y * 0.5) + 3.0) - math.sqrt(y)
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.45)
		tmp = Float64(Float64(Float64(y * 0.5) + 3.0) - sqrt(y));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.45)
		tmp = ((y * 0.5) + 3.0) - sqrt(y);
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.45], N[(N[(N[(y * 0.5), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[y], $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 < 0.450000000000000011

   1. Initial program 97.9%

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

      1. associate-+l+ 97.9%

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

      2. associate-+l+ 97.9%

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

      3. +-commutative 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+l- 63.1%

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

      6. +-commutative 63.1%

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

      7. +-commutative 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in t around inf 55.2%

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

   5. Taylor expanded in x around 0 27.6%

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

   6. Taylor expanded in y around 0 26.1%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 26.1%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   9. Taylor expanded in z around 0 25.2%

      \[\leadsto \color{blue}{\left(3 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 0.450000000000000011 < z

   1. Initial program 89.2%

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

      1. associate-+l+ 89.2%

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

      2. associate-+l+ 89.2%

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

      3. +-commutative 89.2%

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

      4. +-commutative 89.2%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in t around inf 41.5%

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

   5. Taylor expanded in x around 0 5.5%

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

      1. expm1-log1p-u 5.6%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      2. expm1-udef 5.6%

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

      3. log1p-udef 5.6%

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

      4. add-exp-log 5.5%

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

   7. Applied egg-rr 5.5%

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

   8. Taylor expanded in z around inf 29.6%

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

      1. associate--l+ 51.4%

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

   10. Simplified 51.4%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.45:\\ \;\;\;\;\left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 14: 83.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{z + 1} + \left(2 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3e+15)
   (+ (sqrt (+ z 1.0)) (- 2.0 (sqrt z)))
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = sqrt((z + 1.0)) + (2.0 - sqrt(z));
	} else {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3d+15) then
        tmp = sqrt((z + 1.0d0)) + (2.0d0 - sqrt(z))
    else
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+15) {
		tmp = Math.sqrt((z + 1.0)) + (2.0 - Math.sqrt(z));
	} else {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3e+15:
		tmp = math.sqrt((z + 1.0)) + (2.0 - math.sqrt(z))
	else:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3e+15)
		tmp = Float64(sqrt(Float64(z + 1.0)) + Float64(2.0 - sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3e+15)
		tmp = sqrt((z + 1.0)) + (2.0 - sqrt(z));
	else
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3e+15], N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[(2.0 - 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 < 3e15

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. associate-+l+ 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+l- 63.2%

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

      6. +-commutative 63.2%

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

      7. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in t around inf 55.0%

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

   5. Taylor expanded in x around 0 27.3%

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

   6. Taylor expanded in y around 0 45.0%

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

      1. +-commutative 45.0%

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

      2. associate--l+ 45.0%

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

   8. Simplified 45.0%

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

   if 3e15 < z

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-+l+ 89.3%

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

      3. +-commutative 89.3%

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

      4. +-commutative 89.3%

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

      5. associate-+l- 89.3%

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

      6. +-commutative 89.3%

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

      7. +-commutative 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in t around inf 40.7%

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

   5. Taylor expanded in x around 0 4.3%

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

      1. expm1-log1p-u 4.4%

         \[\leadsto \left(1 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y}\right)\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

      2. expm1-udef 4.4%

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

      3. log1p-udef 4.4%

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

      4. add-exp-log 4.3%

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

   7. Applied egg-rr 4.3%

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

   8. Taylor expanded in z around inf 30.7%

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

      1. associate--l+ 53.5%

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

   10. Simplified 53.5%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{z + 1} + \left(2 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 15: 58.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.72:\\ \;\;\;\;\left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 0.72)
   (- (+ (* y 0.5) 3.0) (sqrt y))
   (+ 2.0 (- (* y 0.5) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.72) {
		tmp = ((y * 0.5) + 3.0) - sqrt(y);
	} else {
		tmp = 2.0 + ((y * 0.5) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 0.72d0) then
        tmp = ((y * 0.5d0) + 3.0d0) - sqrt(y)
    else
        tmp = 2.0d0 + ((y * 0.5d0) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 0.72) {
		tmp = ((y * 0.5) + 3.0) - Math.sqrt(y);
	} else {
		tmp = 2.0 + ((y * 0.5) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 0.72:
		tmp = ((y * 0.5) + 3.0) - math.sqrt(y)
	else:
		tmp = 2.0 + ((y * 0.5) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 0.72)
		tmp = Float64(Float64(Float64(y * 0.5) + 3.0) - sqrt(y));
	else
		tmp = Float64(2.0 + Float64(Float64(y * 0.5) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 0.72)
		tmp = ((y * 0.5) + 3.0) - sqrt(y);
	else
		tmp = 2.0 + ((y * 0.5) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 0.72], N[(N[(N[(y * 0.5), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.71999999999999997

   1. Initial program 97.9%

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

      1. associate-+l+ 97.9%

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

      2. associate-+l+ 97.9%

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

      3. +-commutative 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+l- 63.1%

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

      6. +-commutative 63.1%

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

      7. +-commutative 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in t around inf 55.2%

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

   5. Taylor expanded in x around 0 27.6%

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

   6. Taylor expanded in y around 0 26.1%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 26.1%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   9. Taylor expanded in z around 0 25.2%

      \[\leadsto \color{blue}{\left(3 + 0.5 \cdot y\right) - \sqrt{y}} \]

   if 0.71999999999999997 < z

   1. Initial program 89.2%

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

      1. associate-+l+ 89.2%

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

      2. associate-+l+ 89.2%

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

      3. +-commutative 89.2%

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

      4. +-commutative 89.2%

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

      5. associate-+l- 87.6%

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

      6. +-commutative 87.6%

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

      7. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in t around inf 41.5%

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

   5. Taylor expanded in x around 0 5.5%

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

   6. Taylor expanded in y around 0 6.1%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
   7. Step-by-step derivation

      1. *-commutative 6.1%

         \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   8. Simplified 6.1%

      \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

   9. Taylor expanded in z around inf 29.9%

      \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \sqrt{y}} \]
   10. Step-by-step derivation

       1. associate--l+ 29.9%

          \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \sqrt{y}\right)} \]

       2. *-commutative 29.9%

          \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \sqrt{y}\right) \]

   11. Simplified 29.9%

       \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \sqrt{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.72:\\ \;\;\;\;\left(y \cdot 0.5 + 3\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;2 + \left(y \cdot 0.5 - \sqrt{y}\right)\\ \end{array} \]

Alternative 16: 42.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 2 + \left(y \cdot 0.5 - \sqrt{y}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ 2.0 (- (* y 0.5) (sqrt y))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 2.0 + ((y * 0.5) - sqrt(y));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 2.0d0 + ((y * 0.5d0) - sqrt(y))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 2.0 + ((y * 0.5) - Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 2.0 + ((y * 0.5) - math.sqrt(y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(2.0 + Float64(Float64(y * 0.5) - sqrt(y)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 2.0 + ((y * 0.5) - sqrt(y));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(2.0 + N[(N[(y * 0.5), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-+l+ 93.8%

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

   2. associate-+l+ 93.8%

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

   3. +-commutative 93.8%

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

   4. +-commutative 93.8%

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

   5. associate-+l- 74.5%

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

   6. +-commutative 74.5%

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

   7. +-commutative 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in t around inf 48.8%

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

5. Taylor expanded in x around 0 17.4%

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

6. Taylor expanded in y around 0 16.8%

   \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
7. Step-by-step derivation

   1. *-commutative 16.8%

      \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

8. Simplified 16.8%

   \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

9. Taylor expanded in z around inf 21.5%

   \[\leadsto \color{blue}{\left(2 + 0.5 \cdot y\right) - \sqrt{y}} \]
10. Step-by-step derivation

    1. associate--l+ 21.5%

       \[\leadsto \color{blue}{2 + \left(0.5 \cdot y - \sqrt{y}\right)} \]

    2. *-commutative 21.5%

       \[\leadsto 2 + \left(\color{blue}{y \cdot 0.5} - \sqrt{y}\right) \]

11. Simplified 21.5%

    \[\leadsto \color{blue}{2 + \left(y \cdot 0.5 - \sqrt{y}\right)} \]

12. Final simplification 21.5%

    \[\leadsto 2 + \left(y \cdot 0.5 - \sqrt{y}\right) \]

Alternative 17: 6.1% accurate, 274.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ y \cdot 0.5 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* y 0.5))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return y * 0.5;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * 0.5d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return y * 0.5;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return y * 0.5

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(y * 0.5)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = y * 0.5;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(y * 0.5), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-+l+ 93.8%

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

   2. associate-+l+ 93.8%

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

   3. +-commutative 93.8%

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

   4. +-commutative 93.8%

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

   5. associate-+l- 74.5%

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

   6. +-commutative 74.5%

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

   7. +-commutative 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in t around inf 48.8%

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

5. Taylor expanded in x around 0 17.4%

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

6. Taylor expanded in y around 0 16.8%

   \[\leadsto \left(1 + \left(\color{blue}{\left(1 + 0.5 \cdot y\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
7. Step-by-step derivation

   1. *-commutative 16.8%

      \[\leadsto \left(1 + \left(\left(1 + \color{blue}{y \cdot 0.5}\right) + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

8. Simplified 16.8%

   \[\leadsto \left(1 + \left(\color{blue}{\left(1 + y \cdot 0.5\right)} + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

9. Taylor expanded in y around inf 5.3%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
10. Step-by-step derivation

    1. *-commutative 5.3%

       \[\leadsto \color{blue}{y \cdot 0.5} \]

11. Simplified 5.3%

    \[\leadsto \color{blue}{y \cdot 0.5} \]

12. Final simplification 5.3%

    \[\leadsto y \cdot 0.5 \]

Developer target: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023332 
(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))))