Main:z from

Percentage Accurate: 91.7% → 99.2%

Time: 37.5s

Alternatives: 24

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% 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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 - \sqrt{y} \leq 0:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(\left(1 - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y} + t\_1}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \end{array} \]

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 (+ y 1.0))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= (- t_1 (sqrt y)) 0.0)
     (+
      (+
       (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (* 0.5 (sqrt (/ 1.0 y))))
       (* 0.5 (sqrt (/ 1.0 z))))
      t_2)
     (+
      t_2
      (+
       (+ (- 1.0 (sqrt x)) (/ (- (+ y 1.0) y) (+ (sqrt y) t_1)))
       (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if ((t_1 - sqrt(y)) <= 0.0) {
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + t_2;
	} else {
		tmp = t_2 + (((1.0 - sqrt(x)) + (((y + 1.0) - y) / (sqrt(y) + t_1))) + (1.0 / (sqrt(z) + sqrt((1.0 + 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((y + 1.0d0))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if ((t_1 - sqrt(y)) <= 0.0d0) then
        tmp = (((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (0.5d0 * sqrt((1.0d0 / y)))) + (0.5d0 * sqrt((1.0d0 / z)))) + t_2
    else
        tmp = t_2 + (((1.0d0 - sqrt(x)) + (((y + 1.0d0) - y) / (sqrt(y) + t_1))) + (1.0d0 / (sqrt(z) + sqrt((1.0d0 + 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((y + 1.0));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if ((t_1 - Math.sqrt(y)) <= 0.0) {
		tmp = (((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (0.5 * Math.sqrt((1.0 / y)))) + (0.5 * Math.sqrt((1.0 / z)))) + t_2;
	} else {
		tmp = t_2 + (((1.0 - Math.sqrt(x)) + (((y + 1.0) - y) / (Math.sqrt(y) + t_1))) + (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if (t_1 - math.sqrt(y)) <= 0.0:
		tmp = (((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (0.5 * math.sqrt((1.0 / y)))) + (0.5 * math.sqrt((1.0 / z)))) + t_2
	else:
		tmp = t_2 + (((1.0 - math.sqrt(x)) + (((y + 1.0) - y) / (math.sqrt(y) + t_1))) + (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_1 - sqrt(y)) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + t_2);
	else
		tmp = Float64(t_2 + Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(Float64(Float64(y + 1.0) - y) / Float64(sqrt(y) + t_1))) + Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + 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((y + 1.0));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if ((t_1 - sqrt(y)) <= 0.0)
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + t_2;
	else
		tmp = t_2 + (((1.0 - sqrt(x)) + (((y + 1.0) - y) / (sqrt(y) + t_1))) + (1.0 / (sqrt(z) + sqrt((1.0 + 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(t$95$2 + N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + 1.0), $MachinePrecision] - y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf 46.4%

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

   4. Taylor expanded in y around inf 51.6%

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

      1. flip-- 51.8%

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

      2. add-sqr-sqrt 43.5%

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

      3. +-commutative 43.5%

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

      4. add-sqr-sqrt 51.8%

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

      5. +-commutative 51.8%

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

   6. Applied egg-rr 51.8%

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

      1. associate--l+ 53.4%

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

      2. +-inverses 53.4%

         \[\leadsto \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval 53.4%

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

      4. +-commutative 53.4%

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

   8. Simplified 53.4%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 96.5%

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

      1. flip-- 96.7%

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

      2. add-sqr-sqrt 79.5%

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

      3. +-commutative 79.5%

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

      4. add-sqr-sqrt 96.7%

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

      5. +-commutative 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. associate-+r- 97.2%

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

      2. +-inverses 97.2%

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

      3. metadata-eval 97.2%

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

      4. +-commutative 97.2%

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

   6. Simplified 97.2%

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

      1. flip-- 97.4%

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

      2. add-sqr-sqrt 97.7%

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

      3. +-commutative 97.7%

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

      4. add-sqr-sqrt 98.2%

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

      5. +-commutative 98.2%

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

      6. +-commutative 98.2%

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

   8. Applied egg-rr 98.2%

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

   9. Taylor expanded in x around 0 57.4%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(1 - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{y + 1}\\ t_3 := \left(t\_2 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_1 + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 1.9999999995:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \mathsf{fma}\left(0.5, x, \frac{1}{\sqrt{y} + t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(t\_1 + \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (+ (- t_2 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x)))))
   (if (<= t_3 5e-7)
     (+
      t_1
      (+
       (* 0.5 (sqrt (/ 1.0 z)))
       (+ (* 0.5 (sqrt (/ 1.0 y))) (* 0.5 (sqrt (/ 1.0 x))))))
     (if (<= t_3 1.9999999995)
       (+ (- 1.0 (sqrt x)) (fma 0.5 x (/ 1.0 (+ (sqrt y) t_2))))
       (+
        2.0
        (- (+ t_1 (sqrt (+ 1.0 z))) (+ (sqrt z) (+ (sqrt y) (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((y + 1.0));
	double t_3 = (t_2 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x));
	double tmp;
	if (t_3 <= 5e-7) {
		tmp = t_1 + ((0.5 * sqrt((1.0 / z))) + ((0.5 * sqrt((1.0 / y))) + (0.5 * sqrt((1.0 / x)))));
	} else if (t_3 <= 1.9999999995) {
		tmp = (1.0 - sqrt(x)) + fma(0.5, x, (1.0 / (sqrt(y) + t_2)));
	} else {
		tmp = 2.0 + ((t_1 + sqrt((1.0 + z))) - (sqrt(z) + (sqrt(y) + sqrt(x))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = Float64(Float64(t_2 - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x)))
	tmp = 0.0
	if (t_3 <= 5e-7)
		tmp = Float64(t_1 + Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(0.5 * sqrt(Float64(1.0 / x))))));
	elseif (t_3 <= 1.9999999995)
		tmp = Float64(Float64(1.0 - sqrt(x)) + fma(0.5, x, Float64(1.0 / Float64(sqrt(y) + t_2))));
	else
		tmp = Float64(2.0 + Float64(Float64(t_1 + sqrt(Float64(1.0 + z))) - Float64(sqrt(z) + Float64(sqrt(y) + sqrt(x)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-7], N[(t$95$1 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.9999999995], N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * x + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(t$95$1 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 4.99999999999999977e-7

   1. Initial program 67.9%

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

   3. Taylor expanded in z around inf 37.4%

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

   4. Taylor expanded in y around inf 47.5%

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

   5. Taylor expanded in x around inf 51.3%

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

   if 4.99999999999999977e-7 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.9999999995

   1. Initial program 96.1%

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

      1. associate-+l+ 96.1%

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

      2. +-commutative 96.1%

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

      3. associate-+r- 67.4%

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

      4. associate-+l- 62.2%

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

      5. +-commutative 62.2%

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

      6. +-commutative 62.2%

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

      7. associate--l+ 62.2%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in x around inf 24.4%

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

      1. flip-- 24.4%

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

      2. add-sqr-sqrt 14.6%

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

      3. add-sqr-sqrt 24.6%

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

   7. Applied egg-rr 24.6%

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

      1. associate--l+ 25.1%

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

      2. +-inverses 25.1%

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

      3. metadata-eval 25.1%

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

      4. +-commutative 25.1%

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

   9. Simplified 25.1%

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

   10. Taylor expanded in x around 0 25.2%

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

       1. +-commutative 25.2%

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

       2. associate--l+ 25.2%

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

       3. fma-define 25.2%

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

   12. Simplified 25.2%

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

   if 1.9999999995 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   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. +-commutative 98.1%

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

      3. associate-+r- 98.2%

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

      4. associate-+l- 98.2%

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

      5. +-commutative 98.2%

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

      6. +-commutative 98.2%

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

      7. associate--l+ 98.2%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in y around 0 78.7%

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

   6. Taylor expanded in x around 0 26.8%

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

      1. associate--l+ 56.6%

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

      2. associate--r+ 45.8%

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

      3. +-commutative 45.8%

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

      4. metadata-eval 45.8%

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

      5. rem-square-sqrt 45.8%

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

      6. hypot-undefine 45.8%

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

      7. associate-+r- 78.4%

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

      8. hypot-undefine 78.4%

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

      9. metadata-eval 78.4%

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

      10. rem-square-sqrt 78.4%

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

      11. associate-+r+ 78.4%

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

   8. Simplified 78.4%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(0.5 \cdot \sqrt{\frac{1}{y}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right) \leq 1.9999999995:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \mathsf{fma}\left(0.5, x, \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := t\_1 - \sqrt{y}\\ t_3 := 0.5 \cdot \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_3\right) + t\_4\\ \mathbf{elif}\;t\_2 \leq 0.9999999995:\\ \;\;\;\;t\_4 + \left(t\_3 + \left(\left(1 - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y} + t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(t\_4 + \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (- t_1 (sqrt y)))
        (t_3 (* 0.5 (sqrt (/ 1.0 z))))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_2 0.0)
     (+
      (+
       (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (* 0.5 (sqrt (/ 1.0 y))))
       t_3)
      t_4)
     (if (<= t_2 0.9999999995)
       (+
        t_4
        (+ t_3 (+ (- 1.0 (sqrt x)) (/ (- (+ y 1.0) y) (+ (sqrt y) t_1)))))
       (+
        2.0
        (- (+ t_4 (sqrt (+ 1.0 z))) (+ (sqrt z) (+ (sqrt y) (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = t_1 - sqrt(y);
	double t_3 = 0.5 * sqrt((1.0 / z));
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + t_3) + t_4;
	} else if (t_2 <= 0.9999999995) {
		tmp = t_4 + (t_3 + ((1.0 - sqrt(x)) + (((y + 1.0) - y) / (sqrt(y) + t_1))));
	} else {
		tmp = 2.0 + ((t_4 + sqrt((1.0 + z))) - (sqrt(z) + (sqrt(y) + 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = t_1 - sqrt(y)
    t_3 = 0.5d0 * sqrt((1.0d0 / z))
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_2 <= 0.0d0) then
        tmp = (((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (0.5d0 * sqrt((1.0d0 / y)))) + t_3) + t_4
    else if (t_2 <= 0.9999999995d0) then
        tmp = t_4 + (t_3 + ((1.0d0 - sqrt(x)) + (((y + 1.0d0) - y) / (sqrt(y) + t_1))))
    else
        tmp = 2.0d0 + ((t_4 + sqrt((1.0d0 + z))) - (sqrt(z) + (sqrt(y) + 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((y + 1.0));
	double t_2 = t_1 - Math.sqrt(y);
	double t_3 = 0.5 * Math.sqrt((1.0 / z));
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (0.5 * Math.sqrt((1.0 / y)))) + t_3) + t_4;
	} else if (t_2 <= 0.9999999995) {
		tmp = t_4 + (t_3 + ((1.0 - Math.sqrt(x)) + (((y + 1.0) - y) / (Math.sqrt(y) + t_1))));
	} else {
		tmp = 2.0 + ((t_4 + Math.sqrt((1.0 + z))) - (Math.sqrt(z) + (Math.sqrt(y) + 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((y + 1.0))
	t_2 = t_1 - math.sqrt(y)
	t_3 = 0.5 * math.sqrt((1.0 / z))
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 0.0:
		tmp = (((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (0.5 * math.sqrt((1.0 / y)))) + t_3) + t_4
	elif t_2 <= 0.9999999995:
		tmp = t_4 + (t_3 + ((1.0 - math.sqrt(x)) + (((y + 1.0) - y) / (math.sqrt(y) + t_1))))
	else:
		tmp = 2.0 + ((t_4 + math.sqrt((1.0 + z))) - (math.sqrt(z) + (math.sqrt(y) + 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(y + 1.0))
	t_2 = Float64(t_1 - sqrt(y))
	t_3 = Float64(0.5 * sqrt(Float64(1.0 / z)))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + t_3) + t_4);
	elseif (t_2 <= 0.9999999995)
		tmp = Float64(t_4 + Float64(t_3 + Float64(Float64(1.0 - sqrt(x)) + Float64(Float64(Float64(y + 1.0) - y) / Float64(sqrt(y) + t_1)))));
	else
		tmp = Float64(2.0 + Float64(Float64(t_4 + sqrt(Float64(1.0 + z))) - Float64(sqrt(z) + Float64(sqrt(y) + 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((y + 1.0));
	t_2 = t_1 - sqrt(y);
	t_3 = 0.5 * sqrt((1.0 / z));
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + t_3) + t_4;
	elseif (t_2 <= 0.9999999995)
		tmp = t_4 + (t_3 + ((1.0 - sqrt(x)) + (((y + 1.0) - y) / (sqrt(y) + t_1))));
	else
		tmp = 2.0 + ((t_4 + sqrt((1.0 + z))) - (sqrt(z) + (sqrt(y) + 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999995], N[(t$95$4 + N[(t$95$3 + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + 1.0), $MachinePrecision] - y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(N[(t$95$4 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.0

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf 46.4%

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

   4. Taylor expanded in y around inf 51.6%

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

      1. flip-- 51.8%

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

      2. add-sqr-sqrt 43.5%

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

      3. +-commutative 43.5%

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

      4. add-sqr-sqrt 51.8%

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

      5. +-commutative 51.8%

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

   6. Applied egg-rr 51.8%

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

      1. associate--l+ 53.4%

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

      2. +-inverses 53.4%

         \[\leadsto \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval 53.4%

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

      4. +-commutative 53.4%

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

   8. Simplified 53.4%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.99999999949999996

   1. Initial program 88.6%

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

   3. Taylor expanded in z around inf 48.0%

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

   4. Taylor expanded in x around 0 40.3%

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

      1. flip-- 90.4%

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

      2. add-sqr-sqrt 92.8%

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

      3. +-commutative 92.8%

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

      4. add-sqr-sqrt 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   6. Applied egg-rr 42.7%

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

   if 0.99999999949999996 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 97.6%

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

      1. 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. +-commutative 97.6%

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

      3. associate-+r- 64.8%

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

      4. associate-+l- 59.4%

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

      5. +-commutative 59.4%

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

      6. +-commutative 59.4%

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

      7. associate--l+ 59.5%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y around 0 47.9%

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

   6. Taylor expanded in x around 0 15.5%

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

      1. associate--l+ 38.1%

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

      2. associate--r+ 29.8%

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

      3. +-commutative 29.8%

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

      4. metadata-eval 29.8%

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

      5. rem-square-sqrt 29.8%

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

      6. hypot-undefine 29.8%

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

      7. associate-+r- 48.3%

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

      8. hypot-undefine 48.3%

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

      9. metadata-eval 48.3%

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

      10. rem-square-sqrt 48.3%

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

      11. associate-+r+ 48.3%

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

   8. Simplified 48.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{elif}\;\sqrt{y + 1} - \sqrt{y} \leq 0.9999999995:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\left(1 - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \left(\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(t\_1 + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ y 1.0)) (sqrt y)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 5e-5)
     (+
      (+
       (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (* 0.5 (sqrt (/ 1.0 y))))
       (* 0.5 (sqrt (/ 1.0 z))))
      t_2)
     (+
      t_2
      (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) (+ t_1 (- 1.0 (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + t_2;
	} else {
		tmp = t_2 + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (t_1 + (1.0 - 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((y + 1.0d0)) - sqrt(y)
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (t_1 <= 5d-5) then
        tmp = (((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (0.5d0 * sqrt((1.0d0 / y)))) + (0.5d0 * sqrt((1.0d0 / z)))) + t_2
    else
        tmp = t_2 + ((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + (t_1 + (1.0d0 - 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((y + 1.0)) - Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (0.5 * Math.sqrt((1.0 / y)))) + (0.5 * Math.sqrt((1.0 / z)))) + t_2;
	} else {
		tmp = t_2 + ((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + (t_1 + (1.0 - 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((y + 1.0)) - math.sqrt(y)
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if t_1 <= 5e-5:
		tmp = (((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (0.5 * math.sqrt((1.0 / y)))) + (0.5 * math.sqrt((1.0 / z)))) + t_2
	else:
		tmp = t_2 + ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + (t_1 + (1.0 - math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + t_2);
	else
		tmp = Float64(t_2 + Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(t_1 + Float64(1.0 - 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((y + 1.0)) - sqrt(y);
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + t_2;
	else
		tmp = t_2 + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (t_1 + (1.0 - sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 5.00000000000000024e-5

   1. Initial program 83.2%

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

   3. Taylor expanded in z around inf 45.5%

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

   4. Taylor expanded in y around inf 50.7%

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

      1. flip-- 50.9%

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

      2. add-sqr-sqrt 42.8%

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

      3. +-commutative 42.8%

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

      4. add-sqr-sqrt 50.9%

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

      5. +-commutative 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. associate--l+ 52.4%

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

      2. +-inverses 52.4%

         \[\leadsto \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval 52.4%

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

      4. +-commutative 52.4%

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

   8. Simplified 52.4%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 97.6%

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

      2. add-sqr-sqrt 79.8%

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

      3. +-commutative 79.8%

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

      4. add-sqr-sqrt 97.7%

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

      5. +-commutative 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate-+r- 98.2%

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

      2. +-inverses 98.2%

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

      3. metadata-eval 98.2%

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

      4. +-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around 0 57.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t}\\ \mathbf{if}\;y \leq 1.1:\\ \;\;\;\;t\_1 + \left(\left(1 - \sqrt{y}\right) + \left(\left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + t\_2}\right) - \sqrt{x}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + t\_1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(t\_2 - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ 1.0 t))))
   (if (<= y 1.1)
     (+
      t_1
      (+
       (- 1.0 (sqrt y))
       (-
        (- (+ (sqrt (+ 1.0 z)) (/ 1.0 (+ (sqrt t) t_2))) (sqrt x))
        (sqrt z))))
     (+
      (+
       (+ (/ 1.0 (+ (sqrt x) t_1)) (* 0.5 (sqrt (/ 1.0 y))))
       (* 0.5 (sqrt (/ 1.0 z))))
      (- t_2 (sqrt t))))))

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 + t));
	double tmp;
	if (y <= 1.1) {
		tmp = t_1 + ((1.0 - sqrt(y)) + (((sqrt((1.0 + z)) + (1.0 / (sqrt(t) + t_2))) - sqrt(x)) - sqrt(z)));
	} else {
		tmp = (((1.0 / (sqrt(x) + t_1)) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + (t_2 - sqrt(t));
	}
	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 + t))
    if (y <= 1.1d0) then
        tmp = t_1 + ((1.0d0 - sqrt(y)) + (((sqrt((1.0d0 + z)) + (1.0d0 / (sqrt(t) + t_2))) - sqrt(x)) - sqrt(z)))
    else
        tmp = (((1.0d0 / (sqrt(x) + t_1)) + (0.5d0 * sqrt((1.0d0 / y)))) + (0.5d0 * sqrt((1.0d0 / z)))) + (t_2 - sqrt(t))
    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 + t));
	double tmp;
	if (y <= 1.1) {
		tmp = t_1 + ((1.0 - Math.sqrt(y)) + (((Math.sqrt((1.0 + z)) + (1.0 / (Math.sqrt(t) + t_2))) - Math.sqrt(x)) - Math.sqrt(z)));
	} else {
		tmp = (((1.0 / (Math.sqrt(x) + t_1)) + (0.5 * Math.sqrt((1.0 / y)))) + (0.5 * Math.sqrt((1.0 / z)))) + (t_2 - Math.sqrt(t));
	}
	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 + t))
	tmp = 0
	if y <= 1.1:
		tmp = t_1 + ((1.0 - math.sqrt(y)) + (((math.sqrt((1.0 + z)) + (1.0 / (math.sqrt(t) + t_2))) - math.sqrt(x)) - math.sqrt(z)))
	else:
		tmp = (((1.0 / (math.sqrt(x) + t_1)) + (0.5 * math.sqrt((1.0 / y)))) + (0.5 * math.sqrt((1.0 / z)))) + (t_2 - math.sqrt(t))
	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 + t))
	tmp = 0.0
	if (y <= 1.1)
		tmp = Float64(t_1 + Float64(Float64(1.0 - sqrt(y)) + Float64(Float64(Float64(sqrt(Float64(1.0 + z)) + Float64(1.0 / Float64(sqrt(t) + t_2))) - sqrt(x)) - sqrt(z))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + Float64(t_2 - sqrt(t)));
	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 + t));
	tmp = 0.0;
	if (y <= 1.1)
		tmp = t_1 + ((1.0 - sqrt(y)) + (((sqrt((1.0 + z)) + (1.0 / (sqrt(t) + t_2))) - sqrt(x)) - sqrt(z)));
	else
		tmp = (((1.0 / (sqrt(x) + t_1)) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + (t_2 - sqrt(t));
	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 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 1.1], N[(t$95$1 + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1000000000000001

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y around 0 47.9%

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

      1. flip-- 47.9%

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

      2. add-sqr-sqrt 36.8%

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

      3. add-sqr-sqrt 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. associate--l+ 48.2%

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

      2. +-inverses 48.2%

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

      3. metadata-eval 48.2%

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

      4. +-commutative 48.2%

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

   9. Simplified 48.2%

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

   if 1.1000000000000001 < y

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf 44.6%

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

   4. Taylor expanded in y around inf 49.6%

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

      1. flip-- 49.7%

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

      2. add-sqr-sqrt 42.0%

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

      3. +-commutative 42.0%

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

      4. add-sqr-sqrt 49.7%

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

      5. +-commutative 49.7%

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

   6. Applied egg-rr 49.7%

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

      1. associate--l+ 51.2%

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

      2. +-inverses 51.2%

         \[\leadsto \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval 51.2%

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

      4. +-commutative 51.2%

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

   8. Simplified 51.2%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(1 - \sqrt{y}\right) + \left(\left(\left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right) - \sqrt{x}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t} - \sqrt{t}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 0.86:\\ \;\;\;\;t\_2 + \left(\left(1 - \sqrt{y}\right) + \left(\left(\left(t\_1 + \sqrt{1 + z}\right) - \sqrt{x}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + t\_2} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 t)) (sqrt t))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= y 0.86)
     (+
      t_2
      (+ (- 1.0 (sqrt y)) (- (- (+ t_1 (sqrt (+ 1.0 z))) (sqrt x)) (sqrt z))))
     (+
      (+
       (+ (/ 1.0 (+ (sqrt x) t_2)) (* 0.5 (sqrt (/ 1.0 y))))
       (* 0.5 (sqrt (/ 1.0 z))))
      t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t)) - sqrt(t);
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (y <= 0.86) {
		tmp = t_2 + ((1.0 - sqrt(y)) + (((t_1 + sqrt((1.0 + z))) - sqrt(x)) - sqrt(z)));
	} else {
		tmp = (((1.0 / (sqrt(x) + t_2)) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + t)) - sqrt(t)
    t_2 = sqrt((1.0d0 + x))
    if (y <= 0.86d0) then
        tmp = t_2 + ((1.0d0 - sqrt(y)) + (((t_1 + sqrt((1.0d0 + z))) - sqrt(x)) - sqrt(z)))
    else
        tmp = (((1.0d0 / (sqrt(x) + t_2)) + (0.5d0 * sqrt((1.0d0 / y)))) + (0.5d0 * sqrt((1.0d0 / z)))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 0.86) {
		tmp = t_2 + ((1.0 - Math.sqrt(y)) + (((t_1 + Math.sqrt((1.0 + z))) - Math.sqrt(x)) - Math.sqrt(z)));
	} else {
		tmp = (((1.0 / (Math.sqrt(x) + t_2)) + (0.5 * Math.sqrt((1.0 / y)))) + (0.5 * Math.sqrt((1.0 / z)))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 0.86:
		tmp = t_2 + ((1.0 - math.sqrt(y)) + (((t_1 + math.sqrt((1.0 + z))) - math.sqrt(x)) - math.sqrt(z)))
	else:
		tmp = (((1.0 / (math.sqrt(x) + t_2)) + (0.5 * math.sqrt((1.0 / y)))) + (0.5 * math.sqrt((1.0 / z)))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 0.86)
		tmp = Float64(t_2 + Float64(Float64(1.0 - sqrt(y)) + Float64(Float64(Float64(t_1 + sqrt(Float64(1.0 + z))) - sqrt(x)) - sqrt(z))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + t)) - sqrt(t);
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 0.86)
		tmp = t_2 + ((1.0 - sqrt(y)) + (((t_1 + sqrt((1.0 + z))) - sqrt(x)) - sqrt(z)));
	else
		tmp = (((1.0 / (sqrt(x) + t_2)) + (0.5 * sqrt((1.0 / y)))) + (0.5 * sqrt((1.0 / z)))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 0.86], N[(t$95$2 + N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$1 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.859999999999999987

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y around 0 47.9%

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

   if 0.859999999999999987 < y

   1. Initial program 83.6%

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

   3. Taylor expanded in z around inf 44.6%

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

   4. Taylor expanded in y around inf 49.6%

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

      1. flip-- 49.7%

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

      2. add-sqr-sqrt 42.0%

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

      3. +-commutative 42.0%

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

      4. add-sqr-sqrt 49.7%

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

      5. +-commutative 49.7%

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

   6. Applied egg-rr 49.7%

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

      1. associate--l+ 51.2%

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

      2. +-inverses 51.2%

         \[\leadsto \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval 51.2%

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

      4. +-commutative 51.2%

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

   8. Simplified 51.2%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.86:\\ \;\;\;\;\sqrt{1 + x} + \left(\left(1 - \sqrt{y}\right) + \left(\left(\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \sqrt{1 + z}\right) - \sqrt{x}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{z}}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-19}:\\ \;\;\;\;2 + \left(\left(t\_2 + \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 14000000:\\ \;\;\;\;\left(t\_1 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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 (* 0.5 (sqrt (/ 1.0 z)))) (t_2 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= y 4.8e-19)
     (+ 2.0 (- (+ t_2 (sqrt (+ 1.0 z))) (+ (sqrt z) (+ (sqrt y) (sqrt x)))))
     (if (<= y 14000000.0)
       (+
        (+ t_1 (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- 1.0 (sqrt x))))
        (* 0.5 (sqrt (/ 1.0 t))))
       (+
        (+
         (+ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (* 0.5 (sqrt (/ 1.0 y))))
         t_1)
        t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / z));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (y <= 4.8e-19) {
		tmp = 2.0 + ((t_2 + sqrt((1.0 + z))) - (sqrt(z) + (sqrt(y) + sqrt(x))));
	} else if (y <= 14000000.0) {
		tmp = (t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 - sqrt(x)))) + (0.5 * sqrt((1.0 / t)));
	} else {
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + t_1) + t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 * sqrt((1.0d0 / z))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    if (y <= 4.8d-19) then
        tmp = 2.0d0 + ((t_2 + sqrt((1.0d0 + z))) - (sqrt(z) + (sqrt(y) + sqrt(x))))
    else if (y <= 14000000.0d0) then
        tmp = (t_1 + ((sqrt((y + 1.0d0)) - sqrt(y)) + (1.0d0 - sqrt(x)))) + (0.5d0 * sqrt((1.0d0 / t)))
    else
        tmp = (((1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) + (0.5d0 * sqrt((1.0d0 / y)))) + t_1) + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * Math.sqrt((1.0 / z));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double tmp;
	if (y <= 4.8e-19) {
		tmp = 2.0 + ((t_2 + Math.sqrt((1.0 + z))) - (Math.sqrt(z) + (Math.sqrt(y) + Math.sqrt(x))));
	} else if (y <= 14000000.0) {
		tmp = (t_1 + ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (1.0 - Math.sqrt(x)))) + (0.5 * Math.sqrt((1.0 / t)));
	} else {
		tmp = (((1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) + (0.5 * Math.sqrt((1.0 / y)))) + t_1) + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 0.5 * math.sqrt((1.0 / z))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	tmp = 0
	if y <= 4.8e-19:
		tmp = 2.0 + ((t_2 + math.sqrt((1.0 + z))) - (math.sqrt(z) + (math.sqrt(y) + math.sqrt(x))))
	elif y <= 14000000.0:
		tmp = (t_1 + ((math.sqrt((y + 1.0)) - math.sqrt(y)) + (1.0 - math.sqrt(x)))) + (0.5 * math.sqrt((1.0 / t)))
	else:
		tmp = (((1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) + (0.5 * math.sqrt((1.0 / y)))) + t_1) + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / z)))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (y <= 4.8e-19)
		tmp = Float64(2.0 + Float64(Float64(t_2 + sqrt(Float64(1.0 + z))) - Float64(sqrt(z) + Float64(sqrt(y) + sqrt(x)))));
	elseif (y <= 14000000.0)
		tmp = Float64(Float64(t_1 + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(1.0 - sqrt(x)))) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + t_1) + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * sqrt((1.0 / z));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	tmp = 0.0;
	if (y <= 4.8e-19)
		tmp = 2.0 + ((t_2 + sqrt((1.0 + z))) - (sqrt(z) + (sqrt(y) + sqrt(x))));
	elseif (y <= 14000000.0)
		tmp = (t_1 + ((sqrt((y + 1.0)) - sqrt(y)) + (1.0 - sqrt(x)))) + (0.5 * sqrt((1.0 / t)));
	else
		tmp = (((1.0 / (sqrt(x) + sqrt((1.0 + x)))) + (0.5 * sqrt((1.0 / y)))) + t_1) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.8e-19], N[(2.0 + N[(N[(t$95$2 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 14000000.0], N[(N[(t$95$1 + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.80000000000000046e-19

   1. Initial program 97.6%

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

      1. 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. +-commutative 97.6%

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

      3. associate-+r- 64.8%

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

      4. associate-+l- 59.4%

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

      5. +-commutative 59.4%

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

      6. +-commutative 59.4%

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

      7. associate--l+ 59.5%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y around 0 47.9%

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

   6. Taylor expanded in x around 0 15.5%

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

      1. associate--l+ 38.1%

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

      2. associate--r+ 29.8%

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

      3. +-commutative 29.8%

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

      4. metadata-eval 29.8%

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

      5. rem-square-sqrt 29.8%

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

      6. hypot-undefine 29.8%

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

      7. associate-+r- 48.3%

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

      8. hypot-undefine 48.3%

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

      9. metadata-eval 48.3%

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

      10. rem-square-sqrt 48.3%

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

      11. associate-+r+ 48.3%

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

   8. Simplified 48.3%

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

   if 4.80000000000000046e-19 < y < 1.4e7

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf 60.8%

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

   4. Taylor expanded in x around 0 51.0%

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

   5. Taylor expanded in t around inf 25.2%

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

   if 1.4e7 < y

   1. Initial program 83.2%

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

   3. Taylor expanded in z around inf 45.5%

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

   4. Taylor expanded in y around inf 50.7%

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

      1. flip-- 50.9%

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

      2. add-sqr-sqrt 42.8%

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

      3. +-commutative 42.8%

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

      4. add-sqr-sqrt 50.9%

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

      5. +-commutative 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. associate--l+ 52.4%

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

      2. +-inverses 52.4%

         \[\leadsto \left(\left(\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval 52.4%

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

      4. +-commutative 52.4%

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

   8. Simplified 52.4%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-19}:\\ \;\;\;\;2 + \left(\left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \sqrt{1 + z}\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 14000000:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.25 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{1 + x} + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;\left(1 + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 4.25e-19)
   (+
    (sqrt (+ 1.0 x))
    (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt z) (+ (sqrt y) (sqrt x))))))
   (if (<= y 8.5e+27)
     (- (+ 1.0 (/ 1.0 (+ (sqrt y) (sqrt (+ y 1.0))))) (sqrt x))
     (/ 1.0 (+ (sqrt x) (hypot 1.0 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.25e-19) {
		tmp = sqrt((1.0 + x)) + (1.0 + (sqrt((1.0 + z)) - (sqrt(z) + (sqrt(y) + sqrt(x)))));
	} else if (y <= 8.5e+27) {
		tmp = (1.0 + (1.0 / (sqrt(y) + sqrt((y + 1.0))))) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.25e-19) {
		tmp = Math.sqrt((1.0 + x)) + (1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(z) + (Math.sqrt(y) + Math.sqrt(x)))));
	} else if (y <= 8.5e+27) {
		tmp = (1.0 + (1.0 / (Math.sqrt(y) + Math.sqrt((y + 1.0))))) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 4.25e-19:
		tmp = math.sqrt((1.0 + x)) + (1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(z) + (math.sqrt(y) + math.sqrt(x)))))
	elif y <= 8.5e+27:
		tmp = (1.0 + (1.0 / (math.sqrt(y) + math.sqrt((y + 1.0))))) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt(x) + math.hypot(1.0, math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.25e-19)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(z) + Float64(sqrt(y) + sqrt(x))))));
	elseif (y <= 8.5e+27)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(y + 1.0))))) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + hypot(1.0, sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.25e-19)
		tmp = sqrt((1.0 + x)) + (1.0 + (sqrt((1.0 + z)) - (sqrt(z) + (sqrt(y) + sqrt(x)))));
	elseif (y <= 8.5e+27)
		tmp = (1.0 + (1.0 / (sqrt(y) + sqrt((y + 1.0))))) - sqrt(x);
	else
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 4.25e-19], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+27], N[(N[(1.0 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.25000000000000002e-19

   1. Initial program 97.6%

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

      1. 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. +-commutative 97.6%

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

      3. associate-+r- 64.8%

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

      4. associate-+l- 59.4%

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

      5. +-commutative 59.4%

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

      6. +-commutative 59.4%

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

      7. associate--l+ 59.5%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y around 0 47.9%

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

   6. Taylor expanded in t around inf 28.3%

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

      1. associate--l+ 37.5%

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

      2. associate-+r+ 37.5%

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

      3. +-commutative 37.5%

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

   8. Simplified 37.5%

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

   if 4.25000000000000002e-19 < y < 8.5e27

   1. Initial program 75.9%

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

      1. associate-+l+ 75.9%

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

      2. +-commutative 75.9%

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

      3. associate-+r- 68.9%

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

      4. associate-+l- 60.2%

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

      5. +-commutative 60.2%

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

      6. +-commutative 60.2%

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

      7. associate--l+ 60.2%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in x around inf 26.2%

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

      1. flip-- 26.1%

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

      2. add-sqr-sqrt 26.2%

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

      3. add-sqr-sqrt 27.1%

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

   7. Applied egg-rr 27.1%

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

      1. associate--l+ 29.8%

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

      2. +-inverses 29.8%

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

      3. metadata-eval 29.8%

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

      4. +-commutative 29.8%

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

   9. Simplified 29.8%

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

   10. Taylor expanded in x around 0 28.8%

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

   if 8.5e27 < y

   1. Initial program 85.9%

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

      1. associate-+l+ 85.9%

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

      2. +-commutative 85.9%

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

      3. associate-+r- 85.9%

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

      4. associate-+l- 56.8%

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

      5. +-commutative 56.8%

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

      6. +-commutative 56.8%

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

      7. associate--l+ 56.8%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in x around inf 21.9%

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

   6. Taylor expanded in y around inf 21.9%

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

      1. flip-- 21.8%

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

      2. add-sqr-sqrt 22.2%

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

      3. add-sqr-sqrt 21.9%

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

      4. div-sub 21.9%

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

   8. Applied egg-rr 21.9%

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

      1. div-sub 21.9%

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

      2. associate--l+ 24.1%

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

      3. +-inverses 24.1%

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

      4. metadata-eval 24.1%

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

      5. +-commutative 24.1%

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

      6. rem-square-sqrt 24.1%

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

      7. hypot-1-def 24.1%

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

   10. Simplified 24.1%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.25 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{1 + x} + \left(1 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;\left(1 + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{y + 1}}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \mathsf{fma}\left(0.5, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(y + 1.0))))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(1.0 - sqrt(x)) + fma(0.5, x, t_1));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_1);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

      7. associate--l+ 97.1%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. flip-- 37.7%

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

      2. add-sqr-sqrt 28.4%

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

      3. add-sqr-sqrt 37.9%

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

   7. Applied egg-rr 37.9%

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

      1. associate--l+ 38.4%

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

      2. +-inverses 38.4%

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

      3. metadata-eval 38.4%

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

      4. +-commutative 38.4%

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

   9. Simplified 38.4%

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

   10. Taylor expanded in x around 0 38.4%

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

       1. +-commutative 38.4%

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

       2. associate--l+ 38.4%

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

       3. fma-define 38.4%

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

   12. Simplified 38.4%

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

   if 1 < x

   1. Initial program 81.3%

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

      1. associate-+l+ 81.3%

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

      2. +-commutative 81.3%

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

      3. associate-+r- 47.2%

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

      4. associate-+l- 9.9%

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

      5. +-commutative 9.9%

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

      6. +-commutative 9.9%

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

      7. associate--l+ 9.9%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in x around inf 4.3%

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

      1. flip-- 4.3%

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

      2. add-sqr-sqrt 4.1%

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

      3. add-sqr-sqrt 4.3%

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

   7. Applied egg-rr 4.3%

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

      1. associate--l+ 4.3%

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

      2. +-inverses 4.3%

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

      3. metadata-eval 4.3%

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

      4. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in x around inf 24.6%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(1 - \sqrt{x}\right) + \mathsf{fma}\left(0.5, x, \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{y + 1}}\\ \mathbf{if}\;x \leq 49000000:\\ \;\;\;\;\sqrt{1 + x} + \left(t\_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt y) (sqrt (+ y 1.0))))))
   (if (<= x 49000000.0)
     (+ (sqrt (+ 1.0 x)) (- t_1 (sqrt x)))
     (+ (* 0.5 (sqrt (/ 1.0 x))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((y + 1.0)));
	double tmp;
	if (x <= 49000000.0) {
		tmp = sqrt((1.0 + x)) + (t_1 - sqrt(x));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (sqrt(y) + sqrt((y + 1.0d0)))
    if (x <= 49000000.0d0) then
        tmp = sqrt((1.0d0 + x)) + (t_1 - sqrt(x))
    else
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt(y) + Math.sqrt((y + 1.0)));
	double tmp;
	if (x <= 49000000.0) {
		tmp = Math.sqrt((1.0 + x)) + (t_1 - Math.sqrt(x));
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt(y) + math.sqrt((y + 1.0)))
	tmp = 0
	if x <= 49000000.0:
		tmp = math.sqrt((1.0 + x)) + (t_1 - math.sqrt(x))
	else:
		tmp = (0.5 * math.sqrt((1.0 / x))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(y + 1.0))))
	tmp = 0.0
	if (x <= 49000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(t_1 - sqrt(x)));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / (sqrt(y) + sqrt((y + 1.0)));
	tmp = 0.0;
	if (x <= 49000000.0)
		tmp = sqrt((1.0 + x)) + (t_1 - sqrt(x));
	else
		tmp = (0.5 * sqrt((1.0 / x))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 49000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9e7

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. +-commutative 97.0%

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

      3. associate-+r- 97.0%

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

      4. associate-+l- 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

      7. associate--l+ 97.0%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in x around inf 37.3%

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

      1. flip-- 37.3%

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

      2. add-sqr-sqrt 28.1%

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

      3. add-sqr-sqrt 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. associate--l+ 37.9%

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

      2. +-inverses 37.9%

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

      3. metadata-eval 37.9%

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

      4. +-commutative 37.9%

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

   9. Simplified 37.9%

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

   if 4.9e7 < x

   1. Initial program 81.0%

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

      1. associate-+l+ 81.0%

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

      2. +-commutative 81.0%

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

      3. associate-+r- 45.9%

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

      4. associate-+l- 7.7%

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

      5. +-commutative 7.7%

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

      6. +-commutative 7.7%

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

      7. associate--l+ 7.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in x around inf 3.9%

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

      1. flip-- 3.9%

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

      2. add-sqr-sqrt 3.8%

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

      3. add-sqr-sqrt 3.9%

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

   7. Applied egg-rr 3.9%

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

      1. associate--l+ 3.9%

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

      2. +-inverses 3.9%

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

      3. metadata-eval 3.9%

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

      4. +-commutative 3.9%

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

   9. Simplified 3.9%

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

   10. Taylor expanded in x around inf 24.8%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 49000000:\\ \;\;\;\;\sqrt{1 + x} + \left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{y + 1}}\\ \mathbf{if}\;x \leq 0.4:\\ \;\;\;\;\left(1 + t\_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt y) (sqrt (+ y 1.0))))))
   (if (<= x 0.4) (- (+ 1.0 t_1) (sqrt x)) (+ (* 0.5 (sqrt (/ 1.0 x))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((y + 1.0)));
	double tmp;
	if (x <= 0.4) {
		tmp = (1.0 + t_1) - sqrt(x);
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) + t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (sqrt(y) + sqrt((y + 1.0d0)))
    if (x <= 0.4d0) then
        tmp = (1.0d0 + t_1) - sqrt(x)
    else
        tmp = (0.5d0 * sqrt((1.0d0 / x))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt(y) + Math.sqrt((y + 1.0)));
	double tmp;
	if (x <= 0.4) {
		tmp = (1.0 + t_1) - Math.sqrt(x);
	} else {
		tmp = (0.5 * Math.sqrt((1.0 / x))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt(y) + math.sqrt((y + 1.0)))
	tmp = 0
	if x <= 0.4:
		tmp = (1.0 + t_1) - math.sqrt(x)
	else:
		tmp = (0.5 * math.sqrt((1.0 / x))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(y + 1.0))))
	tmp = 0.0
	if (x <= 0.4)
		tmp = Float64(Float64(1.0 + t_1) - sqrt(x));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / (sqrt(y) + sqrt((y + 1.0)));
	tmp = 0.0;
	if (x <= 0.4)
		tmp = (1.0 + t_1) - sqrt(x);
	else
		tmp = (0.5 * sqrt((1.0 / x))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.4], N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.40000000000000002

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

      7. associate--l+ 97.1%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. flip-- 37.7%

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

      2. add-sqr-sqrt 28.4%

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

      3. add-sqr-sqrt 37.9%

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

   7. Applied egg-rr 37.9%

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

      1. associate--l+ 38.4%

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

      2. +-inverses 38.4%

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

      3. metadata-eval 38.4%

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

      4. +-commutative 38.4%

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

   9. Simplified 38.4%

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

   10. Taylor expanded in x around 0 37.7%

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

   if 0.40000000000000002 < x

   1. Initial program 81.3%

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

      1. associate-+l+ 81.3%

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

      2. +-commutative 81.3%

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

      3. associate-+r- 47.2%

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

      4. associate-+l- 9.9%

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

      5. +-commutative 9.9%

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

      6. +-commutative 9.9%

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

      7. associate--l+ 9.9%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in x around inf 4.3%

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

      1. flip-- 4.3%

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

      2. add-sqr-sqrt 4.1%

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

      3. add-sqr-sqrt 4.3%

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

   7. Applied egg-rr 4.3%

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

      1. associate--l+ 4.3%

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

      2. +-inverses 4.3%

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

      3. metadata-eval 4.3%

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

      4. +-commutative 4.3%

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

   9. Simplified 4.3%

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

   10. Taylor expanded in x around inf 24.6%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.4:\\ \;\;\;\;\left(1 + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\left(1 + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.55e-15)
   (- (+ 1.0 (/ 1.0 (+ (sqrt y) (sqrt (+ y 1.0))))) (sqrt x))
   (/ 1.0 (+ (sqrt x) (hypot 1.0 (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.55e-15) {
		tmp = (1.0 + (1.0 / (sqrt(y) + sqrt((y + 1.0))))) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.55e-15) {
		tmp = (1.0 + (1.0 / (Math.sqrt(y) + Math.sqrt((y + 1.0))))) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.55e-15:
		tmp = (1.0 + (1.0 / (math.sqrt(y) + math.sqrt((y + 1.0))))) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt(x) + math.hypot(1.0, math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.55e-15)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(y + 1.0))))) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + hypot(1.0, sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.55e-15)
		tmp = (1.0 + (1.0 / (sqrt(y) + sqrt((y + 1.0))))) - sqrt(x);
	else
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.55e-15], N[(N[(1.0 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5499999999999999e-15

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. +-commutative 97.0%

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

      3. associate-+r- 97.0%

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

      4. associate-+l- 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

      7. associate--l+ 97.0%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in x around inf 36.8%

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

      1. flip-- 36.7%

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

      2. add-sqr-sqrt 27.9%

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

      3. add-sqr-sqrt 36.9%

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

   7. Applied egg-rr 36.9%

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

      1. associate--l+ 37.4%

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

      2. +-inverses 37.4%

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

      3. metadata-eval 37.4%

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

      4. +-commutative 37.4%

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

   9. Simplified 37.4%

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

   10. Taylor expanded in x around 0 37.4%

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

   if 1.5499999999999999e-15 < x

   1. Initial program 82.2%

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

      1. associate-+l+ 82.2%

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

      2. +-commutative 82.2%

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

      3. associate-+r- 49.8%

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

      4. associate-+l- 14.4%

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

      5. +-commutative 14.4%

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

      6. +-commutative 14.4%

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

      7. associate--l+ 14.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in x around inf 7.0%

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

   6. Taylor expanded in y around inf 6.5%

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

      1. flip-- 6.5%

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

      2. add-sqr-sqrt 7.1%

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

      3. add-sqr-sqrt 6.5%

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

      4. div-sub 6.5%

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

   8. Applied egg-rr 6.5%

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

      1. div-sub 6.5%

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

      2. associate--l+ 10.1%

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

      3. +-inverses 10.1%

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

      4. metadata-eval 10.1%

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

      5. +-commutative 10.1%

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

      6. rem-square-sqrt 10.1%

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

      7. hypot-1-def 10.1%

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

   10. Simplified 10.1%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\left(1 + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.1e-25)
   (+ 1.0 (- (sqrt (+ y 1.0)) (+ (sqrt y) (sqrt x))))
   (/ 1.0 (+ (sqrt x) (hypot 1.0 (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.1e-25) {
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.1e-25) {
		tmp = 1.0 + (Math.sqrt((y + 1.0)) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 3.1e-25:
		tmp = 1.0 + (math.sqrt((y + 1.0)) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = 1.0 / (math.sqrt(x) + math.hypot(1.0, math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.1e-25)
		tmp = Float64(1.0 + Float64(sqrt(Float64(y + 1.0)) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + hypot(1.0, sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 3.1e-25)
		tmp = 1.0 + (sqrt((y + 1.0)) - (sqrt(y) + sqrt(x)));
	else
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 3.1e-25], N[(1.0 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.09999999999999995e-25

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. +-commutative 96.9%

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

      3. associate-+r- 96.9%

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

      4. associate-+l- 96.9%

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

      5. +-commutative 96.9%

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

      6. +-commutative 96.9%

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

      7. associate--l+ 96.9%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around inf 37.6%

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

   6. Taylor expanded in x around 0 23.7%

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

      1. +-commutative 23.7%

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

      2. associate--r+ 23.2%

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

      3. associate-+r- 37.6%

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

      4. associate-+r- 37.6%

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

      5. associate--r+ 37.6%

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

      6. metadata-eval 37.6%

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

      7. rem-square-sqrt 37.6%

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

      8. hypot-undefine 37.6%

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

   8. Simplified 37.6%

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

      1. +-commutative 37.6%

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

      2. hypot-undefine 37.6%

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

      3. metadata-eval 37.6%

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

      4. add-sqr-sqrt 37.6%

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

      5. +-commutative 37.6%

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

   10. Applied egg-rr 37.6%

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

   if 3.09999999999999995e-25 < x

   1. Initial program 83.0%

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

      1. associate-+l+ 83.0%

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

      2. +-commutative 83.0%

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

      3. associate-+r- 52.2%

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

      4. associate-+l- 18.5%

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

      5. +-commutative 18.5%

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

      6. +-commutative 18.5%

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

      7. associate--l+ 18.5%

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

   3. Simplified 14.4%

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

   5. Taylor expanded in x around inf 7.6%

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

   6. Taylor expanded in y around inf 7.1%

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

      1. flip-- 7.0%

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

      2. add-sqr-sqrt 7.6%

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

      3. add-sqr-sqrt 7.1%

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

      4. div-sub 7.1%

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

   8. Applied egg-rr 7.1%

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

      1. div-sub 7.1%

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

      2. associate--l+ 10.5%

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

      3. +-inverses 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. rem-square-sqrt 10.5%

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

      7. hypot-1-def 10.5%

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

   10. Simplified 10.5%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;1 + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+19}:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3e+19)
   (+ 1.0 (- (hypot 1.0 (sqrt y)) (sqrt y)))
   (/ 1.0 (+ (sqrt x) (hypot 1.0 (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3e+19) {
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3e+19) {
		tmp = 1.0 + (Math.hypot(1.0, Math.sqrt(y)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3e+19:
		tmp = 1.0 + (math.hypot(1.0, math.sqrt(y)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt(x) + math.hypot(1.0, math.sqrt(x)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3e+19)
		tmp = Float64(1.0 + Float64(hypot(1.0, sqrt(y)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(x) + hypot(1.0, sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3e+19)
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt(x) + hypot(1.0, sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3e+19], N[(1.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e19

   1. Initial program 96.1%

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

      1. associate-+l+ 96.1%

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

      2. +-commutative 96.1%

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

      3. associate-+r- 66.2%

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

      4. associate-+l- 60.4%

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

      5. +-commutative 60.4%

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

      6. +-commutative 60.4%

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

      7. associate--l+ 60.5%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in x around inf 24.4%

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

   6. Taylor expanded in x around 0 23.0%

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

      1. +-commutative 23.0%

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

      2. associate--r+ 23.0%

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

      3. associate-+r- 23.0%

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

      4. associate-+r- 23.0%

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

      5. associate--r+ 23.0%

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

      6. metadata-eval 23.0%

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

      7. rem-square-sqrt 23.0%

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

      8. hypot-undefine 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in y around inf 45.6%

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

   if 3e19 < y

   1. Initial program 83.9%

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

      1. associate-+l+ 83.9%

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

      2. +-commutative 83.9%

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

      3. associate-+r- 83.9%

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

      4. associate-+l- 56.1%

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

      5. +-commutative 56.1%

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

      6. +-commutative 56.1%

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

      7. associate--l+ 56.1%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in x around inf 21.2%

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

   6. Taylor expanded in y around inf 21.2%

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

      1. flip-- 21.1%

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

      2. add-sqr-sqrt 21.5%

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

      3. add-sqr-sqrt 21.2%

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

      4. div-sub 21.2%

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

   8. Applied egg-rr 21.2%

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

      1. div-sub 21.2%

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

      2. associate--l+ 23.4%

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

      3. +-inverses 23.4%

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

      4. metadata-eval 23.4%

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

      5. +-commutative 23.4%

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

      6. rem-square-sqrt 23.4%

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

      7. hypot-1-def 23.4%

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

   10. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5:\\ \;\;\;\;1 + \left(\mathsf{hypot}\left(1, \sqrt{y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 8.5)
   (+ 1.0 (- (hypot 1.0 (sqrt y)) (sqrt y)))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 8.5) {
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 8.5) {
		tmp = 1.0 + (Math.hypot(1.0, Math.sqrt(y)) - Math.sqrt(y));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 8.5:
		tmp = 1.0 + (math.hypot(1.0, math.sqrt(y)) - math.sqrt(y))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 8.5)
		tmp = Float64(1.0 + Float64(hypot(1.0, sqrt(y)) - sqrt(y)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 8.5)
		tmp = 1.0 + (hypot(1.0, sqrt(y)) - sqrt(y));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 8.5], N[(1.0 + N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[y], $MachinePrecision] ^ 2], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.5

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. +-commutative 97.1%

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

      3. associate-+r- 97.1%

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

      4. associate-+l- 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

      7. associate--l+ 97.1%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in x around inf 37.6%

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

   6. Taylor expanded in x around 0 22.3%

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

      1. +-commutative 22.3%

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

      2. associate--r+ 21.8%

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

      3. associate-+r- 36.9%

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

      4. associate-+r- 36.9%

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

      5. associate--r+ 36.4%

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

      6. metadata-eval 36.4%

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

      7. rem-square-sqrt 36.4%

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

      8. hypot-undefine 36.4%

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

   8. Simplified 36.4%

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

   9. Taylor expanded in y around inf 36.4%

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

   if 8.5 < x

   1. Initial program 81.1%

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

      1. associate-+l+ 81.1%

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

      2. +-commutative 81.1%

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

      3. associate-+r- 46.7%

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

      4. associate-+l- 9.2%

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

      5. +-commutative 9.2%

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

      6. +-commutative 9.2%

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

      7. associate--l+ 9.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around inf 4.2%

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

   6. Taylor expanded in y around inf 3.7%

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

   7. Taylor expanded in x around inf 7.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 64.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around inf 24.2%

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

   6. Taylor expanded in x around 0 22.7%

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

      1. +-commutative 22.7%

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

      2. associate--r+ 22.7%

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

      3. associate-+r- 22.7%

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

      4. associate-+r- 22.7%

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

      5. associate--r+ 22.7%

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

      6. metadata-eval 22.7%

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

      7. rem-square-sqrt 22.7%

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

      8. hypot-undefine 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in y around 0 22.7%

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

   if 1 < y

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. +-commutative 83.6%

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

      3. associate-+r- 82.9%

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

      4. associate-+l- 56.6%

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

      5. +-commutative 56.6%

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

      6. +-commutative 56.6%

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

      7. associate--l+ 56.6%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in x around inf 21.7%

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

   6. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

      2. associate--r+ 4.2%

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

      3. associate-+r- 19.8%

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

      4. associate-+r- 19.8%

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

      5. associate--r+ 27.3%

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

      6. metadata-eval 27.3%

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

      7. rem-square-sqrt 27.3%

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

      8. hypot-undefine 27.3%

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

   8. Simplified 27.3%

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

   9. Taylor expanded in y around inf 20.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;1 + \left(\left(1 + y \cdot 0.5\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.0)
   (- (+ 2.0 (* y 0.5)) (+ (sqrt y) (sqrt x)))
   (- (+ 1.0 (* 0.5 (sqrt (/ 1.0 y)))) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.0) {
		tmp = (2.0 + (y * 0.5)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (1.0 + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = (2.0d0 + (y * 0.5d0)) - (sqrt(y) + sqrt(x))
    else
        tmp = (1.0d0 + (0.5d0 * sqrt((1.0d0 / y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.0) {
		tmp = (2.0 + (y * 0.5)) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (1.0 + (0.5 * Math.sqrt((1.0 / y)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 1.0:
		tmp = (2.0 + (y * 0.5)) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (1.0 + (0.5 * math.sqrt((1.0 / y)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(Float64(2.0 + Float64(y * 0.5)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * sqrt(Float64(1.0 / y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = (2.0 + (y * 0.5)) - (sqrt(y) + sqrt(x));
	else
		tmp = (1.0 + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 1.0], N[(N[(2.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around inf 24.2%

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

   6. Taylor expanded in x around 0 22.7%

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

      1. +-commutative 22.7%

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

      2. associate--r+ 22.7%

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

      3. associate-+r- 22.7%

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

      4. associate-+r- 22.7%

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

      5. associate--r+ 22.7%

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

      6. metadata-eval 22.7%

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

      7. rem-square-sqrt 22.7%

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

      8. hypot-undefine 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in y around 0 22.7%

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

   if 1 < y

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. +-commutative 83.6%

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

      3. associate-+r- 82.9%

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

      4. associate-+l- 56.6%

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

      5. +-commutative 56.6%

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

      6. +-commutative 56.6%

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

      7. associate--l+ 56.6%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in x around inf 21.7%

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

   6. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

      2. associate--r+ 4.2%

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

      3. associate-+r- 19.8%

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

      4. associate-+r- 19.8%

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

      5. associate--r+ 27.3%

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

      6. metadata-eval 27.3%

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

      7. rem-square-sqrt 27.3%

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

      8. hypot-undefine 27.3%

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

   8. Simplified 27.3%

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

   9. Taylor expanded in y around inf 20.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\left(2 + y \cdot 0.5\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.37:\\ \;\;\;\;1 + \left(1 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.37)
   (+ 1.0 (- 1.0 (+ (sqrt y) (sqrt x))))
   (- (+ 1.0 (* 0.5 (sqrt (/ 1.0 y)))) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.37) {
		tmp = 1.0 + (1.0 - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.37d0) then
        tmp = 1.0d0 + (1.0d0 - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (0.5d0 * sqrt((1.0d0 / y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.37) {
		tmp = 1.0 + (1.0 - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (0.5 * Math.sqrt((1.0 / y)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.37:
		tmp = 1.0 + (1.0 - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (0.5 * math.sqrt((1.0 / y)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.37)
		tmp = Float64(1.0 + Float64(1.0 - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * sqrt(Float64(1.0 / y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.37)
		tmp = 1.0 + (1.0 - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (0.5 * sqrt((1.0 / y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.37], N[(1.0 + N[(1.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.37

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around inf 24.2%

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

   6. Taylor expanded in x around 0 22.7%

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

      1. +-commutative 22.7%

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

      2. associate--r+ 22.7%

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

      3. associate-+r- 22.7%

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

      4. associate-+r- 22.7%

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

      5. associate--r+ 22.7%

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

      6. metadata-eval 22.7%

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

      7. rem-square-sqrt 22.7%

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

      8. hypot-undefine 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in y around 0 22.7%

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

   if 0.37 < y

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. +-commutative 83.6%

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

      3. associate-+r- 82.9%

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

      4. associate-+l- 56.6%

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

      5. +-commutative 56.6%

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

      6. +-commutative 56.6%

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

      7. associate--l+ 56.6%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in x around inf 21.7%

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

   6. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

      2. associate--r+ 4.2%

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

      3. associate-+r- 19.8%

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

      4. associate-+r- 19.8%

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

      5. associate--r+ 27.3%

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

      6. metadata-eval 27.3%

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

      7. rem-square-sqrt 27.3%

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

      8. hypot-undefine 27.3%

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

   8. Simplified 27.3%

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

   9. Taylor expanded in y around inf 20.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.37:\\ \;\;\;\;1 + \left(1 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;1 + \left(1 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.85)
   (+ 1.0 (- 1.0 (+ (sqrt y) (sqrt x))))
   (- (+ 1.0 (* x 0.5)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.85) {
		tmp = 1.0 + (1.0 - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.85d0) then
        tmp = 1.0d0 + (1.0d0 - (sqrt(y) + sqrt(x)))
    else
        tmp = (1.0d0 + (x * 0.5d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.85) {
		tmp = 1.0 + (1.0 - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = (1.0 + (x * 0.5)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.85:
		tmp = 1.0 + (1.0 - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = (1.0 + (x * 0.5)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.85)
		tmp = Float64(1.0 + Float64(1.0 - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.85)
		tmp = 1.0 + (1.0 - (sqrt(y) + sqrt(x)));
	else
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.85], N[(1.0 + N[(1.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.849999999999999978

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around inf 24.2%

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

   6. Taylor expanded in x around 0 22.7%

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

      1. +-commutative 22.7%

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

      2. associate--r+ 22.7%

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

      3. associate-+r- 22.7%

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

      4. associate-+r- 22.7%

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

      5. associate--r+ 22.7%

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

      6. metadata-eval 22.7%

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

      7. rem-square-sqrt 22.7%

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

      8. hypot-undefine 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in y around 0 22.7%

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

   if 0.849999999999999978 < y

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. +-commutative 83.6%

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

      3. associate-+r- 82.9%

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

      4. associate-+l- 56.6%

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

      5. +-commutative 56.6%

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

      6. +-commutative 56.6%

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

      7. associate--l+ 56.6%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in x around inf 21.7%

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

   6. Taylor expanded in y around inf 21.4%

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

   7. Taylor expanded in x around 0 22.0%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right) - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;1 + \left(1 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;2 - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.85) (- 2.0 (+ (sqrt y) (sqrt x))) (- (+ 1.0 (* x 0.5)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.85) {
		tmp = 2.0 - (sqrt(y) + sqrt(x));
	} else {
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 0.85d0) then
        tmp = 2.0d0 - (sqrt(y) + sqrt(x))
    else
        tmp = (1.0d0 + (x * 0.5d0)) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.85) {
		tmp = 2.0 - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (1.0 + (x * 0.5)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 0.85:
		tmp = 2.0 - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (1.0 + (x * 0.5)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.85)
		tmp = Float64(2.0 - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 0.85)
		tmp = 2.0 - (sqrt(y) + sqrt(x));
	else
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 0.85], N[(2.0 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.849999999999999978

   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+ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-+r- 65.6%

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

      4. associate-+l- 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate--l+ 60.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in x around inf 24.2%

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

   6. Taylor expanded in x around 0 22.7%

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

      1. +-commutative 22.7%

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

      2. associate--r+ 22.7%

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

      3. associate-+r- 22.7%

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

      4. associate-+r- 22.7%

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

      5. associate--r+ 22.7%

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

      6. metadata-eval 22.7%

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

      7. rem-square-sqrt 22.7%

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

      8. hypot-undefine 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in y around 0 22.7%

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

   if 0.849999999999999978 < y

   1. Initial program 83.6%

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

      1. associate-+l+ 83.6%

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

      2. +-commutative 83.6%

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

      3. associate-+r- 82.9%

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

      4. associate-+l- 56.6%

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

      5. +-commutative 56.6%

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

      6. +-commutative 56.6%

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

      7. associate--l+ 56.6%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in x around inf 21.7%

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

   6. Taylor expanded in y around inf 21.4%

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

   7. Taylor expanded in x around 0 22.0%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right) - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;2 - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 40.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.0], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 97.1%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 97.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 37.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   6. Taylor expanded in y around inf 27.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   7. Taylor expanded in x around 0 27.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right) - \sqrt{x}} \]

   if 1 < x

   1. Initial program 81.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 81.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 81.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 47.2%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 9.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 9.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 9.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 9.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 9.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 4.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   6. Taylor expanded in y around inf 3.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   7. Taylor expanded in x around inf 7.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 39.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 0.21) (- 1.0 (sqrt x)) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.21) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 0.21d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.21) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 0.21:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 0.21)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 0.21)
		tmp = 1.0 - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 0.21], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.209999999999999992

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 97.1%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 97.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 97.1%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 97.1%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 97.1%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 37.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   6. Taylor expanded in y around inf 27.6%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   7. Taylor expanded in x around 0 27.0%

      \[\leadsto \color{blue}{1 - \sqrt{x}} \]

   if 0.209999999999999992 < x

   1. Initial program 81.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Step-by-step derivation

      1. associate-+l+ 81.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

      2. +-commutative 81.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      3. associate-+r- 47.2%

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

      4. associate-+l- 9.9%

         \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

      5. +-commutative 9.9%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

      6. +-commutative 9.9%

         \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

      7. associate--l+ 9.9%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

   3. Simplified 9.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 4.3%

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

   6. Taylor expanded in y around inf 3.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

   7. Taylor expanded in x around inf 7.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 34.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 90.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 90.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 74.9%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 58.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 58.3%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 58.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 45.8%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 22.8%

   \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) \]

6. Taylor expanded in y around inf 17.0%

   \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

7. Taylor expanded in x around 0 15.6%

   \[\leadsto \color{blue}{1 - \sqrt{x}} \]
8. Add Preprocessing

Alternative 24: 1.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -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 = -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 -Math.sqrt(y);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(y)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(y))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -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[Sqrt[y], $MachinePrecision])

Derivation

1. Initial program 90.1%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Step-by-step derivation

   1. associate-+l+ 90.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)} \]

   2. +-commutative 90.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   3. associate-+r- 74.9%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right) \]

   4. associate-+l- 58.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \sqrt{x + 1}\right) - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right)} \]

   5. +-commutative 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} - \left(\sqrt{x} - \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\right)\right) \]

   6. +-commutative 58.3%

      \[\leadsto \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \left(\sqrt{x} - \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}\right) \]

   7. associate--l+ 58.3%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{x} - \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right)\right)} \]

3. Simplified 45.8%

   \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 23.0%

   \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(1 - \sqrt{y}\right)} - \left(\left(\sqrt{x} - \left(\sqrt{1 + z} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right) + \sqrt{z}\right)\right) \]

6. Taylor expanded in y around inf 1.6%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{y}} \]
7. Step-by-step derivation

   1. neg-mul-1 1.6%

      \[\leadsto \color{blue}{-\sqrt{y}} \]

8. Simplified 1.6%

   \[\leadsto \color{blue}{-\sqrt{y}} \]
9. Add Preprocessing

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 2024096 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (+ (+ (+ (/ 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))))