Main:z from

Percentage Accurate: 91.8% → 99.3%

Time: 51.0s

Alternatives: 20

Speedup: 1.6×

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: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{x + 1}\\ \mathbf{if}\;\left(\left(\left(t\_5 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + t\_3\right) + t\_4 \leq 0.99:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_5} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + t\_1}\right) + \left(\frac{1}{\sqrt{z} + t\_2} + t\_4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (sqrt (+ x 1.0))))
   (if (<= (+ (+ (+ (- t_5 (sqrt x)) (- t_1 (sqrt y))) t_3) t_4) 0.99)
     (+ (+ (/ 1.0 (+ (sqrt x) t_5)) (* 0.5 (sqrt (/ 1.0 y)))) t_3)
     (+
      (+
       (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
       (/ 1.0 (+ (sqrt y) t_1)))
      (+ (/ 1.0 (+ (sqrt z) t_2)) t_4)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt((x + 1.0));
	double tmp;
	if (((((t_5 - sqrt(x)) + (t_1 - sqrt(y))) + t_3) + t_4) <= 0.99) {
		tmp = ((1.0 / (sqrt(x) + t_5)) + (0.5 * sqrt((1.0 / y)))) + t_3;
	} else {
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x))) + (1.0 / (sqrt(y) + t_1))) + ((1.0 / (sqrt(z) + t_2)) + t_4);
	}
	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) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((1.0d0 + t)) - sqrt(t)
    t_5 = sqrt((x + 1.0d0))
    if (((((t_5 - sqrt(x)) + (t_1 - sqrt(y))) + t_3) + t_4) <= 0.99d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_5)) + (0.5d0 * sqrt((1.0d0 / y)))) + t_3
    else
        tmp = ((1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))) + (1.0d0 / (sqrt(y) + t_1))) + ((1.0d0 / (sqrt(z) + t_2)) + t_4)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_5 = Math.sqrt((x + 1.0));
	double tmp;
	if (((((t_5 - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + t_3) + t_4) <= 0.99) {
		tmp = ((1.0 / (Math.sqrt(x) + t_5)) + (0.5 * Math.sqrt((1.0 / y)))) + t_3;
	} else {
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x))) + (1.0 / (Math.sqrt(y) + t_1))) + ((1.0 / (Math.sqrt(z) + t_2)) + t_4);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_5 = math.sqrt((x + 1.0))
	tmp = 0
	if ((((t_5 - math.sqrt(x)) + (t_1 - math.sqrt(y))) + t_3) + t_4) <= 0.99:
		tmp = ((1.0 / (math.sqrt(x) + t_5)) + (0.5 * math.sqrt((1.0 / y)))) + t_3
	else:
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))) + (1.0 / (math.sqrt(y) + t_1))) + ((1.0 / (math.sqrt(z) + t_2)) + t_4)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t_5 - sqrt(x)) + Float64(t_1 - sqrt(y))) + t_3) + t_4) <= 0.99)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_5)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + t_3);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x))) + Float64(1.0 / Float64(sqrt(y) + t_1))) + Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + t_4));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((1.0 + t)) - sqrt(t);
	t_5 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((((t_5 - sqrt(x)) + (t_1 - sqrt(y))) + t_3) + t_4) <= 0.99)
		tmp = ((1.0 / (sqrt(x) + t_5)) + (0.5 * sqrt((1.0 / y)))) + t_3;
	else
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x))) + (1.0 / (sqrt(y) + t_1))) + ((1.0 / (sqrt(z) + t_2)) + t_4);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision], 0.99], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.98999999999999999

   1. Initial program 22.6%

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

      1. associate-+l+ 22.6%

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

      2. sub-neg 22.6%

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

      3. sub-neg 22.6%

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

      4. +-commutative 22.6%

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

      5. +-commutative 22.6%

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

      6. +-commutative 22.6%

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

   3. Simplified 22.6%

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

      1. flip-- 22.6%

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

      2. div-inv 22.6%

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

      3. add-sqr-sqrt 8.6%

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

      4. +-commutative 8.6%

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

      5. add-sqr-sqrt 22.6%

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

      6. +-commutative 22.6%

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

   6. Applied egg-rr 22.6%

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

      1. associate--l+ 46.0%

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

      2. +-inverses 46.0%

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

      3. metadata-eval 46.0%

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

      4. *-lft-identity 46.0%

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

      5. +-commutative 46.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in t around inf 41.7%

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

   10. Taylor expanded in y around inf 56.0%

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

   if 0.98999999999999999 < (+.f64 (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. sub-neg 96.2%

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

      3. sub-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

      1. flip-- 96.3%

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

      2. div-inv 96.3%

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

      3. add-sqr-sqrt 80.1%

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

      4. add-sqr-sqrt 96.4%

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

      5. associate--l+ 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. +-inverses 97.2%

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

      2. metadata-eval 97.2%

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

      3. *-lft-identity 97.2%

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

      4. +-commutative 97.2%

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

   8. Simplified 97.2%

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

      1. flip-- 97.4%

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

      2. div-inv 97.4%

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

      3. add-sqr-sqrt 78.8%

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

      4. add-sqr-sqrt 98.1%

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

      5. associate--l+ 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. +-inverses 98.7%

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

       2. metadata-eval 98.7%

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

       3. *-lft-identity 98.7%

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

       4. +-commutative 98.7%

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

   12. Simplified 98.7%

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

   13. Taylor expanded in x around 0 52.2%

       \[\leadsto \left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   14. Step-by-step derivation

       1. associate--l+ 52.2%

          \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   15. Simplified 52.2%

       \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \leq 0.99:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{1 + z}\\ \mathbf{if}\;t\_1 - \sqrt{x} \leq 0.99:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_1} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(t\_2 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + t\_2}\\ \end{array} \end{array} \]

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 (sqrt (+ x 1.0))) (t_2 (sqrt (+ 1.0 z))))
   (if (<= (- t_1 (sqrt x)) 0.99)
     (+ (+ (/ 1.0 (+ (sqrt x) t_1)) (* 0.5 (sqrt (/ 1.0 y)))) (- t_2 (sqrt z)))
     (+
      (+
       (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
       (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y)))))
      (/ 1.0 (+ (sqrt z) t_2))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((1.0 + z));
	double tmp;
	if ((t_1 - sqrt(x)) <= 0.99) {
		tmp = ((1.0 / (sqrt(x) + t_1)) + (0.5 * sqrt((1.0 / y)))) + (t_2 - sqrt(z));
	} else {
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x))) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) + (1.0 / (sqrt(z) + t_2));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((1.0d0 + z))
    if ((t_1 - sqrt(x)) <= 0.99d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_1)) + (0.5d0 * sqrt((1.0d0 / y)))) + (t_2 - sqrt(z))
    else
        tmp = ((1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))) + (1.0d0 / (sqrt(y) + sqrt((1.0d0 + y))))) + (1.0d0 / (sqrt(z) + t_2))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((1.0 + z));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 0.99) {
		tmp = ((1.0 / (Math.sqrt(x) + t_1)) + (0.5 * Math.sqrt((1.0 / y)))) + (t_2 - Math.sqrt(z));
	} else {
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x))) + (1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y))))) + (1.0 / (Math.sqrt(z) + t_2));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((1.0 + z))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 0.99:
		tmp = ((1.0 / (math.sqrt(x) + t_1)) + (0.5 * math.sqrt((1.0 / y)))) + (t_2 - math.sqrt(z))
	else:
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))) + (1.0 / (math.sqrt(y) + math.sqrt((1.0 + y))))) + (1.0 / (math.sqrt(z) + t_2))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 0.99)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(t_2 - sqrt(z)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - sqrt(x))) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) + Float64(1.0 / Float64(sqrt(z) + t_2)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((1.0 + z));
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 0.99)
		tmp = ((1.0 / (sqrt(x) + t_1)) + (0.5 * sqrt((1.0 / y)))) + (t_2 - sqrt(z));
	else
		tmp = ((1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x))) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) + (1.0 / (sqrt(z) + t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.99], 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[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.98999999999999999

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. sub-neg 86.6%

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

      3. sub-neg 86.6%

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

      4. +-commutative 86.6%

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

      5. +-commutative 86.6%

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

      6. +-commutative 86.6%

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

   3. Simplified 86.6%

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

      1. flip-- 86.6%

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

      2. div-inv 86.6%

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

      3. add-sqr-sqrt 47.3%

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

      4. +-commutative 47.3%

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

      5. add-sqr-sqrt 86.7%

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

      6. +-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. associate--l+ 90.2%

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

      2. +-inverses 90.2%

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

      3. metadata-eval 90.2%

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

      4. *-lft-identity 90.2%

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

      5. +-commutative 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in t around inf 45.6%

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

   10. Taylor expanded in y around inf 23.6%

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

   if 0.98999999999999999 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 95.8%

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

      1. associate-+l+ 95.8%

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

      2. sub-neg 95.8%

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

      3. sub-neg 95.8%

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

      4. +-commutative 95.8%

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

      5. +-commutative 95.8%

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

      6. +-commutative 95.8%

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

   3. Simplified 95.8%

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

      1. flip-- 95.8%

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

      2. div-inv 95.8%

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

      3. add-sqr-sqrt 78.1%

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

      4. add-sqr-sqrt 96.0%

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

      5. associate--l+ 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. +-inverses 97.2%

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

      2. metadata-eval 97.2%

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

      3. *-lft-identity 97.2%

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

      4. +-commutative 97.2%

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

   8. Simplified 97.2%

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

      1. flip-- 97.6%

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

      2. div-inv 97.6%

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

      3. add-sqr-sqrt 73.9%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate--l+ 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. +-inverses 99.1%

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

       2. metadata-eval 99.1%

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

       3. *-lft-identity 99.1%

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

       4. +-commutative 99.1%

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

   12. Simplified 99.1%

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

   13. Taylor expanded in x around 0 99.1%

       \[\leadsto \left(\color{blue}{\left(\left(1 + x \cdot \left(0.5 + -0.125 \cdot x\right)\right) - \sqrt{x}\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]
   14. Step-by-step derivation

       1. associate--l+ 99.1%

          \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   15. Simplified 99.1%

       \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right)} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

   16. Taylor expanded in t around inf 63.8%

       \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} \]
   17. Step-by-step derivation

       1. +-commutative 63.8%

          \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} \]

   18. Simplified 63.8%

       \[\leadsto \left(\left(1 + \left(x \cdot \left(0.5 + -0.125 \cdot x\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \color{blue}{\frac{1}{\sqrt{1 + z} + \sqrt{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.99:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\right) + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 10500000:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(t\_1 + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_1\\ \end{array} \end{array} \]

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 z)) (sqrt z))))
   (if (<= y 10500000.0)
     (+
      (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
      (+ t_1 (/ 1.0 (+ (sqrt t) (sqrt (+ 1.0 t))))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (* 0.5 (sqrt (/ 1.0 y))))
      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 + z)) - sqrt(z);
	double tmp;
	if (y <= 10500000.0) {
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (t_1 + (1.0 / (sqrt(t) + sqrt((1.0 + t)))));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + 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 = sqrt((1.0d0 + z)) - sqrt(z)
    if (y <= 10500000.0d0) then
        tmp = ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + (t_1 + (1.0d0 / (sqrt(t) + sqrt((1.0d0 + t)))))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (0.5d0 * sqrt((1.0d0 / y)))) + 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 + z)) - Math.sqrt(z);
	double tmp;
	if (y <= 10500000.0) {
		tmp = ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + (t_1 + (1.0 / (Math.sqrt(t) + Math.sqrt((1.0 + t)))));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (0.5 * Math.sqrt((1.0 / y)))) + 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 + z)) - math.sqrt(z)
	tmp = 0
	if y <= 10500000.0:
		tmp = ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + (t_1 + (1.0 / (math.sqrt(t) + math.sqrt((1.0 + t)))))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (0.5 * math.sqrt((1.0 / y)))) + 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 + z)) - sqrt(z))
	tmp = 0.0
	if (y <= 10500000.0)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(t_1 + Float64(1.0 / Float64(sqrt(t) + sqrt(Float64(1.0 + t))))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + 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 + z)) - sqrt(z);
	tmp = 0.0;
	if (y <= 10500000.0)
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (t_1 + (1.0 / (sqrt(t) + sqrt((1.0 + t)))));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + 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 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 10500000.0], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e7

   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. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 39.9%

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

      1. flip-- 40.0%

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

      2. div-inv 40.0%

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

      3. add-sqr-sqrt 31.5%

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

      4. +-commutative 31.5%

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

      5. add-sqr-sqrt 40.0%

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

      6. associate--l+ 40.0%

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

   7. Applied egg-rr 40.0%

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

      1. associate-*r/ 40.0%

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

      2. *-rgt-identity 40.0%

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

      3. associate-+r- 40.0%

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

      4. +-commutative 40.0%

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

      5. associate-+r- 40.2%

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

      6. +-inverses 40.2%

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

      7. metadata-eval 40.2%

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

      8. +-commutative 40.2%

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

   9. Simplified 40.2%

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

   if 1.05e7 < y

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. sub-neg 84.4%

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

      3. sub-neg 84.4%

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

      4. +-commutative 84.4%

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

      5. +-commutative 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

      1. flip-- 84.4%

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

      2. div-inv 84.4%

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

      3. add-sqr-sqrt 70.5%

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

      4. +-commutative 70.5%

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

      5. add-sqr-sqrt 84.4%

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

      6. +-commutative 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate--l+ 87.9%

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

      2. +-inverses 87.9%

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

      3. metadata-eval 87.9%

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

      4. *-lft-identity 87.9%

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

      5. +-commutative 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in t around inf 52.0%

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

   10. Taylor expanded in y around inf 55.2%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10500000:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 23000000:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + t\_1} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(t\_1 - \sqrt{z}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 z))))
   (if (<= y 23000000.0)
     (+
      (+ (/ 1.0 (+ (sqrt z) t_1)) (- (sqrt (+ 1.0 t)) (sqrt t)))
      (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (* 0.5 (sqrt (/ 1.0 y))))
      (- t_1 (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (y <= 23000000.0) {
		tmp = ((1.0 / (sqrt(z) + t_1)) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + (t_1 - sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (y <= 23000000.0d0) then
        tmp = ((1.0d0 / (sqrt(z) + t_1)) + (sqrt((1.0d0 + t)) - sqrt(t))) + ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y)))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (0.5d0 * sqrt((1.0d0 / y)))) + (t_1 - sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 23000000.0) {
		tmp = ((1.0 / (Math.sqrt(z) + t_1)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (0.5 * Math.sqrt((1.0 / y)))) + (t_1 - Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 23000000.0:
		tmp = ((1.0 / (math.sqrt(z) + t_1)) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (0.5 * math.sqrt((1.0 / y)))) + (t_1 - math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 23000000.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(t_1 - sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 23000000.0)
		tmp = ((1.0 / (sqrt(z) + t_1)) + (sqrt((1.0 + t)) - sqrt(t))) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + (t_1 - sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 23000000.0], N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3e7

   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. sub-neg 97.0%

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

      3. sub-neg 97.0%

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

      4. +-commutative 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 40.3%

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

      1. flip-- 97.1%

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

      2. div-inv 97.1%

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

      3. add-sqr-sqrt 80.9%

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

      4. add-sqr-sqrt 97.1%

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

      5. associate--l+ 98.1%

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

   7. Applied egg-rr 40.9%

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

      1. +-inverses 98.1%

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

      2. metadata-eval 98.1%

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

      3. *-lft-identity 98.1%

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

      4. +-commutative 98.1%

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

   9. Simplified 40.9%

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

   if 2.3e7 < y

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. sub-neg 84.4%

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

      3. sub-neg 84.4%

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

      4. +-commutative 84.4%

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

      5. +-commutative 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

      1. flip-- 84.4%

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

      2. div-inv 84.4%

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

      3. add-sqr-sqrt 70.4%

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

      4. +-commutative 70.4%

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

      5. add-sqr-sqrt 84.4%

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

      6. +-commutative 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate--l+ 87.9%

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

      2. +-inverses 87.9%

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

      3. metadata-eval 87.9%

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

      4. *-lft-identity 87.9%

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

      5. +-commutative 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in t around inf 52.3%

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

   10. Taylor expanded in y around inf 55.5%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 23000000:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ \mathbf{if}\;y \leq 14500000:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(t\_1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_1\\ \end{array} \end{array} \]

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 z)) (sqrt z))))
   (if (<= y 14500000.0)
     (+
      (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y)))
      (+ t_1 (- (sqrt (+ 1.0 t)) (sqrt t))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (* 0.5 (sqrt (/ 1.0 y))))
      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 + z)) - sqrt(z);
	double tmp;
	if (y <= 14500000.0) {
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (t_1 + (sqrt((1.0 + t)) - sqrt(t)));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + 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 = sqrt((1.0d0 + z)) - sqrt(z)
    if (y <= 14500000.0d0) then
        tmp = ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y))) + (t_1 + (sqrt((1.0d0 + t)) - sqrt(t)))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (0.5d0 * sqrt((1.0d0 / y)))) + 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 + z)) - Math.sqrt(z);
	double tmp;
	if (y <= 14500000.0) {
		tmp = ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + (t_1 + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (0.5 * Math.sqrt((1.0 / y)))) + 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 + z)) - math.sqrt(z)
	tmp = 0
	if y <= 14500000.0:
		tmp = ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + (t_1 + (math.sqrt((1.0 + t)) - math.sqrt(t)))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (0.5 * math.sqrt((1.0 / y)))) + 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 + z)) - sqrt(z))
	tmp = 0.0
	if (y <= 14500000.0)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(t_1 + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + 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 + z)) - sqrt(z);
	tmp = 0.0;
	if (y <= 14500000.0)
		tmp = ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y))) + (t_1 + (sqrt((1.0 + t)) - sqrt(t)));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + 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 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 14500000.0], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e7

   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. sub-neg 97.0%

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

      3. sub-neg 97.0%

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

      4. +-commutative 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 40.3%

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

   if 1.45e7 < y

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. sub-neg 84.4%

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

      3. sub-neg 84.4%

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

      4. +-commutative 84.4%

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

      5. +-commutative 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

      1. flip-- 84.4%

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

      2. div-inv 84.4%

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

      3. add-sqr-sqrt 70.4%

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

      4. +-commutative 70.4%

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

      5. add-sqr-sqrt 84.4%

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

      6. +-commutative 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate--l+ 87.9%

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

      2. +-inverses 87.9%

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

      3. metadata-eval 87.9%

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

      4. *-lft-identity 87.9%

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

      5. +-commutative 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in t around inf 52.3%

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

   10. Taylor expanded in y around inf 55.5%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14500000:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 27000000:\\ \;\;\;\;\frac{1}{\sqrt{z} + t\_1} + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(t\_1 - \sqrt{z}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 z))))
   (if (<= y 27000000.0)
     (+
      (/ 1.0 (+ (sqrt z) t_1))
      (+ (- 1.0 (sqrt x)) (- (sqrt (+ 1.0 y)) (sqrt y))))
     (+
      (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (* 0.5 (sqrt (/ 1.0 y))))
      (- t_1 (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double tmp;
	if (y <= 27000000.0) {
		tmp = (1.0 / (sqrt(z) + t_1)) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	} else {
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + (t_1 - sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    if (y <= 27000000.0d0) then
        tmp = (1.0d0 / (sqrt(z) + t_1)) + ((1.0d0 - sqrt(x)) + (sqrt((1.0d0 + y)) - sqrt(y)))
    else
        tmp = ((1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (0.5d0 * sqrt((1.0d0 / y)))) + (t_1 - sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double tmp;
	if (y <= 27000000.0) {
		tmp = (1.0 / (Math.sqrt(z) + t_1)) + ((1.0 - Math.sqrt(x)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y)));
	} else {
		tmp = ((1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (0.5 * Math.sqrt((1.0 / y)))) + (t_1 - Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	tmp = 0
	if y <= 27000000.0:
		tmp = (1.0 / (math.sqrt(z) + t_1)) + ((1.0 - math.sqrt(x)) + (math.sqrt((1.0 + y)) - math.sqrt(y)))
	else:
		tmp = ((1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (0.5 * math.sqrt((1.0 / y)))) + (t_1 - math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (y <= 27000000.0)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(t_1 - sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	tmp = 0.0;
	if (y <= 27000000.0)
		tmp = (1.0 / (sqrt(z) + t_1)) + ((1.0 - sqrt(x)) + (sqrt((1.0 + y)) - sqrt(y)));
	else
		tmp = ((1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (0.5 * sqrt((1.0 / y)))) + (t_1 - sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 27000000.0], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e7

   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. sub-neg 97.0%

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

      3. sub-neg 97.0%

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

      4. +-commutative 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 26.0%

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

      1. flip-- 97.1%

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

      2. div-inv 97.1%

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

      3. add-sqr-sqrt 80.9%

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

      4. add-sqr-sqrt 97.1%

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

      5. associate--l+ 98.1%

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

   8. Applied egg-rr 26.5%

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

      1. +-inverses 98.1%

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

      2. metadata-eval 98.1%

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

      3. *-lft-identity 98.1%

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

      4. +-commutative 98.1%

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

   10. Simplified 26.5%

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

   if 2.7e7 < y

   1. Initial program 84.4%

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

      1. associate-+l+ 84.4%

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

      2. sub-neg 84.4%

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

      3. sub-neg 84.4%

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

      4. +-commutative 84.4%

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

      5. +-commutative 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

      1. flip-- 84.4%

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

      2. div-inv 84.4%

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

      3. add-sqr-sqrt 70.4%

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

      4. +-commutative 70.4%

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

      5. add-sqr-sqrt 84.4%

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

      6. +-commutative 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate--l+ 87.9%

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

      2. +-inverses 87.9%

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

      3. metadata-eval 87.9%

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

      4. *-lft-identity 87.9%

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

      5. +-commutative 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in t around inf 52.3%

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

   10. Taylor expanded in y around inf 55.5%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 27000000:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;x \leq 3.75 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(1 - \sqrt{x}\right) + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + t\_1\\ \end{array} \end{array} \]

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 y)) (sqrt y))))
   (if (<= x 3.75e-19)
     (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) (+ (- 1.0 (sqrt x)) t_1))
     (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) 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 + y)) - sqrt(y);
	double tmp;
	if (x <= 3.75e-19) {
		tmp = (1.0 / (sqrt(z) + sqrt((1.0 + z)))) + ((1.0 - sqrt(x)) + t_1);
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + 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 = sqrt((1.0d0 + y)) - sqrt(y)
    if (x <= 3.75d-19) then
        tmp = (1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + ((1.0d0 - sqrt(x)) + t_1)
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + 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 + y)) - Math.sqrt(y);
	double tmp;
	if (x <= 3.75e-19) {
		tmp = (1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + ((1.0 - Math.sqrt(x)) + t_1);
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + 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 + y)) - math.sqrt(y)
	tmp = 0
	if x <= 3.75e-19:
		tmp = (1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + ((1.0 - math.sqrt(x)) + t_1)
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + 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 + y)) - sqrt(y))
	tmp = 0.0
	if (x <= 3.75e-19)
		tmp = Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(Float64(1.0 - sqrt(x)) + t_1));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + 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 + y)) - sqrt(y);
	tmp = 0.0;
	if (x <= 3.75e-19)
		tmp = (1.0 / (sqrt(z) + sqrt((1.0 + z)))) + ((1.0 - sqrt(x)) + t_1);
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + 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 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.75e-19], N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.74999999999999979e-19

   1. Initial program 95.9%

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

      1. associate-+l+ 95.9%

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

      2. sub-neg 95.9%

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

      3. sub-neg 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in x around 0 95.9%

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

   6. Taylor expanded in t around inf 62.1%

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

      1. flip-- 95.9%

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

      2. div-inv 95.9%

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

      3. add-sqr-sqrt 77.2%

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

      4. add-sqr-sqrt 96.1%

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

      5. associate--l+ 97.4%

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

   8. Applied egg-rr 63.2%

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

      1. +-inverses 97.4%

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

      2. metadata-eval 97.4%

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

      3. *-lft-identity 97.4%

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

      4. +-commutative 97.4%

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

   10. Simplified 63.2%

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

   if 3.74999999999999979e-19 < x

   1. Initial program 86.9%

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

      1. associate-+l+ 86.9%

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

      2. sub-neg 86.9%

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

      3. sub-neg 86.9%

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

      4. +-commutative 86.9%

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

      5. +-commutative 86.9%

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

      6. +-commutative 86.9%

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

   3. Simplified 86.9%

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

      1. flip-- 86.9%

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

      2. div-inv 86.9%

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

      3. add-sqr-sqrt 49.3%

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

      4. +-commutative 49.3%

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

      5. add-sqr-sqrt 87.0%

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

      6. +-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. associate--l+ 90.4%

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

      2. +-inverses 90.4%

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

      3. metadata-eval 90.4%

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

      4. *-lft-identity 90.4%

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

      5. +-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in t around inf 46.2%

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

   10. Taylor expanded in z around inf 17.4%

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

       1. +-commutative 17.4%

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

       2. associate-+r- 22.4%

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

       3. +-commutative 22.4%

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

       4. +-commutative 22.4%

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

       5. +-commutative 22.4%

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

   12. Simplified 22.4%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.75 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ \mathbf{if}\;x \leq 3.75 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + t\_1\\ \end{array} \end{array} \]

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 y)) (sqrt y))))
   (if (<= x 3.75e-19)
     (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (+ (- 1.0 (sqrt x)) t_1))
     (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) 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 + y)) - sqrt(y);
	double tmp;
	if (x <= 3.75e-19) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + t_1);
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + 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 = sqrt((1.0d0 + y)) - sqrt(y)
    if (x <= 3.75d-19) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((1.0d0 - sqrt(x)) + t_1)
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + 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 + y)) - Math.sqrt(y);
	double tmp;
	if (x <= 3.75e-19) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((1.0 - Math.sqrt(x)) + t_1);
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + 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 + y)) - math.sqrt(y)
	tmp = 0
	if x <= 3.75e-19:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((1.0 - math.sqrt(x)) + t_1)
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + 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 + y)) - sqrt(y))
	tmp = 0.0
	if (x <= 3.75e-19)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(1.0 - sqrt(x)) + t_1));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + 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 + y)) - sqrt(y);
	tmp = 0.0;
	if (x <= 3.75e-19)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((1.0 - sqrt(x)) + t_1);
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + 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 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.75e-19], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.74999999999999979e-19

   1. Initial program 95.9%

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

      1. associate-+l+ 95.9%

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

      2. sub-neg 95.9%

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

      3. sub-neg 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in x around 0 95.9%

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

   6. Taylor expanded in t around inf 62.1%

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

   if 3.74999999999999979e-19 < x

   1. Initial program 86.9%

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

      1. associate-+l+ 86.9%

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

      2. sub-neg 86.9%

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

      3. sub-neg 86.9%

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

      4. +-commutative 86.9%

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

      5. +-commutative 86.9%

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

      6. +-commutative 86.9%

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

   3. Simplified 86.9%

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

      1. flip-- 86.9%

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

      2. div-inv 86.9%

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

      3. add-sqr-sqrt 49.3%

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

      4. +-commutative 49.3%

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

      5. add-sqr-sqrt 87.0%

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

      6. +-commutative 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. associate--l+ 90.4%

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

      2. +-inverses 90.4%

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

      3. metadata-eval 90.4%

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

      4. *-lft-identity 90.4%

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

      5. +-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in t around inf 46.2%

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

   10. Taylor expanded in z around inf 17.4%

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

       1. +-commutative 17.4%

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

       2. associate-+r- 22.4%

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

       3. +-commutative 22.4%

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

       4. +-commutative 22.4%

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

       5. +-commutative 22.4%

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

   12. Simplified 22.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.75 \cdot 10^{-19}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(1 - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 + y \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.2e-10)
   (+
    (- (sqrt (+ 1.0 z)) (sqrt z))
    (- (+ 2.0 (* y 0.5)) (+ (sqrt x) (sqrt y))))
   (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e-10) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 + (y * 0.5)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.2d-10) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((2.0d0 + (y * 0.5d0)) - (sqrt(x) + sqrt(y)))
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e-10) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((2.0 + (y * 0.5)) - (Math.sqrt(x) + Math.sqrt(y)));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.2e-10:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((2.0 + (y * 0.5)) - (math.sqrt(x) + math.sqrt(y)))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2e-10)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(2.0 + Float64(y * 0.5)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.2e-10)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 + (y * 0.5)) - (sqrt(x) + sqrt(y)));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 2.2e-10], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1999999999999999e-10

   1. Initial program 97.5%

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

      1. associate-+l+ 97.5%

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

      2. sub-neg 97.5%

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

      3. sub-neg 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in x around 0 40.6%

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

   6. Taylor expanded in t around inf 24.9%

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

   7. Taylor expanded in y around 0 24.9%

      \[\leadsto \color{blue}{\left(\left(2 + 0.5 \cdot y\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{1 + z} - \sqrt{z}\right) \]

   if 2.1999999999999999e-10 < y

   1. Initial program 85.5%

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

      1. associate-+l+ 85.5%

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

      2. sub-neg 85.5%

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

      3. sub-neg 85.5%

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

      4. +-commutative 85.5%

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

      5. +-commutative 85.5%

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

      6. +-commutative 85.5%

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

   3. Simplified 85.5%

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

      1. flip-- 85.5%

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

      2. div-inv 85.5%

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

      3. add-sqr-sqrt 69.4%

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

      4. +-commutative 69.4%

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

      5. add-sqr-sqrt 85.5%

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

      6. +-commutative 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. associate--l+ 88.6%

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

      2. +-inverses 88.6%

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

      3. metadata-eval 88.6%

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

      4. *-lft-identity 88.6%

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

      5. +-commutative 88.6%

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

   8. Simplified 88.6%

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

   9. Taylor expanded in t around inf 53.4%

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

   10. Taylor expanded in z around inf 10.6%

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

       1. +-commutative 10.6%

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

       2. associate-+r- 32.2%

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

       3. +-commutative 32.2%

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

       4. +-commutative 32.2%

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

       5. +-commutative 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 + y \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.6e-13)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (- 2.0 (sqrt x)) (sqrt y)))
   (+ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (- (sqrt (+ 1.0 y)) (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.6e-13) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (sqrt((1.0 + y)) - sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 6.6d-13) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((2.0d0 - sqrt(x)) - sqrt(y))
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) + (sqrt((1.0d0 + y)) - sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.6e-13) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((2.0 - Math.sqrt(x)) - Math.sqrt(y));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 6.6e-13:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((2.0 - math.sqrt(x)) - math.sqrt(y))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) + (math.sqrt((1.0 + y)) - math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.6e-13)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(2.0 - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.6e-13)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) + (sqrt((1.0 + y)) - sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 6.6e-13], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6000000000000001e-13

   1. Initial program 97.5%

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

      1. associate-+l+ 97.5%

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

      2. sub-neg 97.5%

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

      3. sub-neg 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in x around 0 40.6%

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

   6. Taylor expanded in t around inf 24.9%

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

   7. Taylor expanded in y around 0 24.9%

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

      1. associate--r+ 24.9%

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

   9. Simplified 24.9%

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

   if 6.6000000000000001e-13 < y

   1. Initial program 85.5%

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

      1. associate-+l+ 85.5%

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

      2. sub-neg 85.5%

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

      3. sub-neg 85.5%

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

      4. +-commutative 85.5%

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

      5. +-commutative 85.5%

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

      6. +-commutative 85.5%

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

   3. Simplified 85.5%

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

      1. flip-- 85.5%

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

      2. div-inv 85.5%

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

      3. add-sqr-sqrt 69.4%

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

      4. +-commutative 69.4%

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

      5. add-sqr-sqrt 85.5%

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

      6. +-commutative 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. associate--l+ 88.6%

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

      2. +-inverses 88.6%

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

      3. metadata-eval 88.6%

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

      4. *-lft-identity 88.6%

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

      5. +-commutative 88.6%

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

   8. Simplified 88.6%

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

   9. Taylor expanded in t around inf 53.4%

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

   10. Taylor expanded in z around inf 10.6%

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

       1. +-commutative 10.6%

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

       2. associate-+r- 32.2%

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

       3. +-commutative 32.2%

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

       4. +-commutative 32.2%

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

       5. +-commutative 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x + 1}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.65:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 0.65)
   (+ (- (sqrt (+ 1.0 z)) (sqrt z)) (- (- 2.0 (sqrt x)) (sqrt y)))
   (+ 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 <= 0.65) {
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	} 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 <= 0.65d0) then
        tmp = (sqrt((1.0d0 + z)) - sqrt(z)) + ((2.0d0 - sqrt(x)) - sqrt(y))
    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 <= 0.65) {
		tmp = (Math.sqrt((1.0 + z)) - Math.sqrt(z)) + ((2.0 - Math.sqrt(x)) - Math.sqrt(y));
	} 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 <= 0.65:
		tmp = (math.sqrt((1.0 + z)) - math.sqrt(z)) + ((2.0 - math.sqrt(x)) - math.sqrt(y))
	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 <= 0.65)
		tmp = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(2.0 - sqrt(x)) - sqrt(y)));
	else
		tmp = Float64(1.0 + Float64(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 <= 0.65)
		tmp = (sqrt((1.0 + z)) - sqrt(z)) + ((2.0 - sqrt(x)) - sqrt(y));
	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, 0.65], N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.650000000000000022

   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. sub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 25.3%

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

   7. Taylor expanded in y around 0 24.3%

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

      1. associate--r+ 24.3%

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

   9. Simplified 24.3%

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

   if 0.650000000000000022 < y

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. sub-neg 84.9%

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

      3. sub-neg 84.9%

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

      4. +-commutative 84.9%

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

      5. +-commutative 84.9%

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

      6. +-commutative 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around 0 53.1%

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

   6. Taylor expanded in t around inf 35.9%

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

   7. Taylor expanded in y around inf 35.5%

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

   8. Taylor expanded in z around inf 25.0%

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

      1. associate--l+ 25.0%

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

   10. Simplified 25.0%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.65:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(2 - \sqrt{x}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.39:\\ \;\;\;\;\left(2 + 0.5 \cdot \sqrt{\frac{1}{t}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 0.39)
   (- (+ 2.0 (* 0.5 (sqrt (/ 1.0 t)))) (+ (sqrt x) (sqrt y)))
   (+ 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 <= 0.39) {
		tmp = (2.0 + (0.5 * sqrt((1.0 / t)))) - (sqrt(x) + sqrt(y));
	} 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 <= 0.39d0) then
        tmp = (2.0d0 + (0.5d0 * sqrt((1.0d0 / t)))) - (sqrt(x) + sqrt(y))
    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 <= 0.39) {
		tmp = (2.0 + (0.5 * Math.sqrt((1.0 / t)))) - (Math.sqrt(x) + Math.sqrt(y));
	} 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 <= 0.39:
		tmp = (2.0 + (0.5 * math.sqrt((1.0 / t)))) - (math.sqrt(x) + math.sqrt(y))
	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 <= 0.39)
		tmp = Float64(Float64(2.0 + Float64(0.5 * sqrt(Float64(1.0 / t)))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(1.0 + Float64(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 <= 0.39)
		tmp = (2.0 + (0.5 * sqrt((1.0 / t)))) - (sqrt(x) + sqrt(y));
	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, 0.39], N[(N[(2.0 + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.39000000000000001

   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. sub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 24.2%

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

   7. Taylor expanded in y around 0 13.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   8. Taylor expanded in z around inf 15.5%

      \[\leadsto \color{blue}{\left(2 + 0.5 \cdot \sqrt{\frac{1}{t}}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 0.39000000000000001 < y

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. sub-neg 84.9%

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

      3. sub-neg 84.9%

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

      4. +-commutative 84.9%

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

      5. +-commutative 84.9%

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

      6. +-commutative 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around 0 53.1%

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

   6. Taylor expanded in t around inf 35.9%

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

   7. Taylor expanded in y around inf 35.5%

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

   8. Taylor expanded in z around inf 25.0%

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

      1. associate--l+ 25.0%

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

   10. Simplified 25.0%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.39:\\ \;\;\;\;\left(2 + 0.5 \cdot \sqrt{\frac{1}{t}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.335:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 0.335)
   (+ 1.0 (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z))))
   (+ 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 <= 0.335) {
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
	} 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 <= 0.335d0) then
        tmp = 1.0d0 + (sqrt((1.0d0 + z)) - (sqrt(x) + sqrt(z)))
    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 <= 0.335) {
		tmp = 1.0 + (Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z)));
	} 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 <= 0.335:
		tmp = 1.0 + (math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z)))
	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 <= 0.335)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + sqrt(z))));
	else
		tmp = Float64(1.0 + Float64(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 <= 0.335)
		tmp = 1.0 + (sqrt((1.0 + z)) - (sqrt(x) + sqrt(z)));
	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, 0.335], N[(1.0 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.33500000000000002

   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. sub-neg 97.6%

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

      3. sub-neg 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

      6. +-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 25.3%

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

   7. Taylor expanded in y around inf 5.3%

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

      1. associate--l+ 11.1%

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

   9. Simplified 11.1%

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

   if 0.33500000000000002 < y

   1. Initial program 84.9%

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

      1. associate-+l+ 84.9%

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

      2. sub-neg 84.9%

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

      3. sub-neg 84.9%

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

      4. +-commutative 84.9%

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

      5. +-commutative 84.9%

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

      6. +-commutative 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around 0 53.1%

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

   6. Taylor expanded in t around inf 35.9%

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

   7. Taylor expanded in y around inf 35.5%

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

   8. Taylor expanded in z around inf 25.0%

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

      1. associate--l+ 25.0%

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

   10. Simplified 25.0%

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

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.335:\\ \;\;\;\;1 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (<= z 3.2e+14)
   (- (sqrt (+ 1.0 z)) (+ (sqrt x) (sqrt z)))
   (+ 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 (z <= 3.2e+14) {
		tmp = sqrt((1.0 + z)) - (sqrt(x) + sqrt(z));
	} 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 (z <= 3.2d+14) then
        tmp = sqrt((1.0d0 + z)) - (sqrt(x) + sqrt(z))
    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 (z <= 3.2e+14) {
		tmp = Math.sqrt((1.0 + z)) - (Math.sqrt(x) + Math.sqrt(z));
	} 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 z <= 3.2e+14:
		tmp = math.sqrt((1.0 + z)) - (math.sqrt(x) + math.sqrt(z))
	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 (z <= 3.2e+14)
		tmp = Float64(sqrt(Float64(1.0 + z)) - Float64(sqrt(x) + sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(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 (z <= 3.2e+14)
		tmp = sqrt((1.0 + z)) - (sqrt(x) + sqrt(z));
	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[z, 3.2e+14], N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.2e14

   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. Step-by-step derivation

      1. associate-+l+ 96.5%

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

      2. sub-neg 96.5%

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

      3. sub-neg 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 26.5%

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

   7. Taylor expanded in x around inf 8.1%

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

      1. mul-1-neg 8.1%

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

   9. Simplified 8.1%

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

   10. Taylor expanded in x around 0 8.1%

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

   if 3.2e14 < z

   1. Initial program 85.1%

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

      1. associate-+l+ 85.1%

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

      2. sub-neg 85.1%

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

      3. sub-neg 85.1%

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

      4. +-commutative 85.1%

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

      5. +-commutative 85.1%

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

      6. +-commutative 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in x around 0 54.1%

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

   6. Taylor expanded in t around inf 35.4%

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

   7. Taylor expanded in y around inf 23.4%

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

   8. Taylor expanded in z around inf 23.4%

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

      1. associate--l+ 23.4%

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

   10. Simplified 23.4%

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

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{1 + z} - \left(\sqrt{x} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.7% accurate, 3.8× speedup

Math

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

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 (<= z 1.0)
   (- 1.0 (+ (sqrt x) (sqrt z)))
   (+ 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 (z <= 1.0) {
		tmp = 1.0 - (sqrt(x) + sqrt(z));
	} 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 (z <= 1.0d0) then
        tmp = 1.0d0 - (sqrt(x) + sqrt(z))
    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 (z <= 1.0) {
		tmp = 1.0 - (Math.sqrt(x) + Math.sqrt(z));
	} 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 z <= 1.0:
		tmp = 1.0 - (math.sqrt(x) + math.sqrt(z))
	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 (z <= 1.0)
		tmp = Float64(1.0 - Float64(sqrt(x) + sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(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 (z <= 1.0)
		tmp = 1.0 - (sqrt(x) + sqrt(z));
	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[z, 1.0], N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   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. sub-neg 97.0%

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

      3. sub-neg 97.0%

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

      4. +-commutative 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 41.0%

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

   6. Taylor expanded in t around inf 26.8%

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

   7. Taylor expanded in x around inf 8.2%

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

      1. mul-1-neg 8.2%

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

   9. Simplified 8.2%

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

   10. Taylor expanded in z around 0 8.2%

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

   if 1 < z

   1. Initial program 85.0%

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

      1. associate-+l+ 85.0%

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

      2. sub-neg 85.0%

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

      3. sub-neg 85.0%

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

      4. +-commutative 85.0%

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

      5. +-commutative 85.0%

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

      6. +-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in x around 0 53.0%

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

   6. Taylor expanded in t around inf 34.9%

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

   7. Taylor expanded in y around inf 23.4%

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

   8. Taylor expanded in z around inf 23.2%

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

      1. associate--l+ 23.2%

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

   10. Simplified 23.2%

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

4. Final simplification 15.6%

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

Alternative 16: 32.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 57000000000000:\\ \;\;\;\;\left(1 + z \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (<= z 57000000000000.0)
   (- (+ 1.0 (* z 0.5)) (+ (sqrt x) (sqrt z)))
   (+ 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 (z <= 57000000000000.0) {
		tmp = (1.0 + (z * 0.5)) - (sqrt(x) + sqrt(z));
	} 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 (z <= 57000000000000.0d0) then
        tmp = (1.0d0 + (z * 0.5d0)) - (sqrt(x) + sqrt(z))
    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 (z <= 57000000000000.0) {
		tmp = (1.0 + (z * 0.5)) - (Math.sqrt(x) + Math.sqrt(z));
	} 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 z <= 57000000000000.0:
		tmp = (1.0 + (z * 0.5)) - (math.sqrt(x) + math.sqrt(z))
	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 (z <= 57000000000000.0)
		tmp = Float64(Float64(1.0 + Float64(z * 0.5)) - Float64(sqrt(x) + sqrt(z)));
	else
		tmp = Float64(1.0 + Float64(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 (z <= 57000000000000.0)
		tmp = (1.0 + (z * 0.5)) - (sqrt(x) + sqrt(z));
	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[z, 57000000000000.0], N[(N[(1.0 + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.7e13

   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. Step-by-step derivation

      1. associate-+l+ 96.5%

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

      2. sub-neg 96.5%

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

      3. sub-neg 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in t around inf 26.5%

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

   7. Taylor expanded in x around inf 8.1%

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

      1. mul-1-neg 8.1%

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

   9. Simplified 8.1%

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

   10. Taylor expanded in z around 0 8.1%

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

       1. *-commutative 8.1%

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

   12. Simplified 8.1%

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

   if 5.7e13 < z

   1. Initial program 85.1%

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

      1. associate-+l+ 85.1%

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

      2. sub-neg 85.1%

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

      3. sub-neg 85.1%

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

      4. +-commutative 85.1%

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

      5. +-commutative 85.1%

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

      6. +-commutative 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in x around 0 54.1%

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

   6. Taylor expanded in t around inf 35.4%

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

   7. Taylor expanded in y around inf 23.4%

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

   8. Taylor expanded in z around inf 23.4%

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

      1. associate--l+ 23.4%

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

   10. Simplified 23.4%

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

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 57000000000000:\\ \;\;\;\;\left(1 + z \cdot 0.5\right) - \left(\sqrt{x} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(0.5 \cdot \sqrt{\frac{1}{y}} - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 11.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   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. sub-neg 97.0%

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

      3. sub-neg 97.0%

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

      4. +-commutative 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 41.0%

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

   6. Taylor expanded in t around inf 26.8%

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

   7. Taylor expanded in x around inf 8.2%

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

      1. mul-1-neg 8.2%

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

   9. Simplified 8.2%

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

   10. Taylor expanded in z around 0 8.2%

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

   if 1 < z

   1. Initial program 85.0%

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

      1. associate-+l+ 85.0%

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

      2. sub-neg 85.0%

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

      3. sub-neg 85.0%

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

      4. +-commutative 85.0%

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

      5. +-commutative 85.0%

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

      6. +-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in x around 0 53.0%

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

   6. Taylor expanded in t around inf 34.9%

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

   7. Taylor expanded in y around inf 23.4%

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

   8. Taylor expanded in y around 0 9.4%

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

4. Final simplification 8.8%

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

Alternative 18: 6.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{t}} \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / t));
}

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 = 0.5d0 * sqrt((1.0d0 / t))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / t));
end

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-+l+ 91.1%

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

   2. sub-neg 91.1%

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

   3. sub-neg 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. +-commutative 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in x around 0 46.9%

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

6. Taylor expanded in t around inf 29.4%

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

7. Taylor expanded in y around 0 7.6%

   \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} + 0.5 \cdot \sqrt{\frac{1}{t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

8. Taylor expanded in t around 0 7.3%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{t}}} \]

9. Final simplification 7.3%

   \[\leadsto 0.5 \cdot \sqrt{\frac{1}{t}} \]
10. Add Preprocessing

Alternative 19: 7.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{y}} \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / 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 = 0.5d0 * sqrt((1.0d0 / y))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / y))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / y)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / 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[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. associate-+l+ 91.1%

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

   2. sub-neg 91.1%

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

   3. sub-neg 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. +-commutative 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in x around 0 46.9%

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

6. Taylor expanded in t around inf 30.8%

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

7. Taylor expanded in y around inf 20.6%

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

8. Taylor expanded in y around 0 8.4%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{y}}} \]

9. Final simplification 8.4%

   \[\leadsto 0.5 \cdot \sqrt{\frac{1}{y}} \]
10. Add Preprocessing

Alternative 20: 1.9% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -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[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 91.1%

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

   1. associate-+l+ 91.1%

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

   2. sub-neg 91.1%

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

   3. sub-neg 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. +-commutative 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in x around 0 46.9%

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

6. Taylor expanded in t around inf 30.8%

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

7. Taylor expanded in x around inf 5.0%

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

   1. mul-1-neg 5.0%

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

9. Simplified 5.0%

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

10. Taylor expanded in x around inf 1.6%

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

    1. neg-mul-1 1.6%

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

12. Simplified 1.6%

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

13. Final simplification 1.6%

    \[\leadsto -\sqrt{x} \]
14. 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 2024071 
(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))))