Main:z from

Percentage Accurate: 92.1% → 97.6%

Time: 17.1s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_2\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + t\_1\right) + t\_2\\ \mathbf{elif}\;t\_3 \leq 3.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{t}, \sqrt{t}, \left(\left(\sqrt{x - -1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y - -1}\right)\right) - \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_2)))
   (if (<= t_3 2e-6)
     (+ (+ (/ 0.5 (* x (sqrt (/ 1.0 x)))) t_1) t_2)
     (if (<= t_3 3.01)
       (fma
        (/ 0.5 t)
        (sqrt t)
        (-
         (- (- (sqrt (- x -1.0)) (sqrt x)) (- (sqrt y) (sqrt (- y -1.0))))
         (/ -1.0 (+ (sqrt (- z -1.0)) (sqrt z)))))
       (+
        (-
         (+ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt (+ 1.0 z))))
         (+ (sqrt x) (+ (sqrt y) (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((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_1) + t_2;
	} else if (t_3 <= 3.01) {
		tmp = fma((0.5 / t), sqrt(t), (((sqrt((x - -1.0)) - sqrt(x)) - (sqrt(y) - sqrt((y - -1.0)))) - (-1.0 / (sqrt((z - -1.0)) + sqrt(z)))));
	} else {
		tmp = ((1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))) + t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_2)
	tmp = 0.0
	if (t_3 <= 2e-6)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) + t_1) + t_2);
	elseif (t_3 <= 3.01)
		tmp = fma(Float64(0.5 / t), sqrt(t), Float64(Float64(Float64(sqrt(Float64(x - -1.0)) - sqrt(x)) - Float64(sqrt(y) - sqrt(Float64(y - -1.0)))) - Float64(-1.0 / Float64(sqrt(Float64(z - -1.0)) + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + z)))) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))) + t_2);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-6], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 3.01], N[(N[(0.5 / t), $MachinePrecision] * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))) < 1.99999999999999991e-6

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

   if 1.99999999999999991e-6 < (+.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))) < 3.0099999999999998

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 88.2

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

   10. Applied rewrites 88.2%

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

   if 3.0099999999999998 < (+.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 92.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 39.7

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

   4. Applied rewrites 39.7%

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

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y - -1}\\ t_2 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;x \leq 135000000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{t - -1}{t}} - 1, \sqrt{t}, \left(\left(\sqrt{x - -1} - \sqrt{x}\right) - \frac{\sqrt{y} \cdot \sqrt{y} - t\_1 \cdot t\_1}{\sqrt{y} + t\_1}\right) - \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5 \cdot \frac{1}{t\_2} - 0.125 \cdot \frac{1}{{x}^{2} \cdot {t\_2}^{3}}}{x} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (- y -1.0))) (t_2 (sqrt (/ 1.0 x))))
   (if (<= x 135000000000.0)
     (fma
      (- (sqrt (/ (- t -1.0) t)) 1.0)
      (sqrt t)
      (-
       (-
        (- (sqrt (- x -1.0)) (sqrt x))
        (/ (- (* (sqrt y) (sqrt y)) (* t_1 t_1)) (+ (sqrt y) t_1)))
       (/ -1.0 (+ (sqrt (- z -1.0)) (sqrt z)))))
     (+
      (+
       (/
        (- (* 0.5 (/ 1.0 t_2)) (* 0.125 (/ 1.0 (* (pow x 2.0) (pow t_2 3.0)))))
        x)
       (- (sqrt (+ z 1.0)) (sqrt z)))
      (- (sqrt (+ t 1.0)) (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y - -1.0));
	double t_2 = sqrt((1.0 / x));
	double tmp;
	if (x <= 135000000000.0) {
		tmp = fma((sqrt(((t - -1.0) / t)) - 1.0), sqrt(t), (((sqrt((x - -1.0)) - sqrt(x)) - (((sqrt(y) * sqrt(y)) - (t_1 * t_1)) / (sqrt(y) + t_1))) - (-1.0 / (sqrt((z - -1.0)) + sqrt(z)))));
	} else {
		tmp = ((((0.5 * (1.0 / t_2)) - (0.125 * (1.0 / (pow(x, 2.0) * pow(t_2, 3.0))))) / x) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y - -1.0))
	t_2 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (x <= 135000000000.0)
		tmp = fma(Float64(sqrt(Float64(Float64(t - -1.0) / t)) - 1.0), sqrt(t), Float64(Float64(Float64(sqrt(Float64(x - -1.0)) - sqrt(x)) - Float64(Float64(Float64(sqrt(y) * sqrt(y)) - Float64(t_1 * t_1)) / Float64(sqrt(y) + t_1))) - Float64(-1.0 / Float64(sqrt(Float64(z - -1.0)) + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / t_2)) - Float64(0.125 * Float64(1.0 / Float64((x ^ 2.0) * (t_2 ^ 3.0))))) / x) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 135000000000.0], N[(N[(N[Sqrt[N[(N[(t - -1.0), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(1.0 / N[(N[Power[x, 2.0], $MachinePrecision] * N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e11

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-+.f64 93.7

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

   9. Applied rewrites 93.7%

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

   if 1.35e11 < x

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\frac{1}{2} \cdot \frac{1}{\sqrt{\frac{1}{x}}} - \frac{1}{8} \cdot \frac{1}{{x}^{2} \cdot {\left(\sqrt{\frac{1}{x}}\right)}^{3}}}{\color{blue}{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{1}{2} \cdot \frac{1}{\sqrt{\frac{1}{x}}} - \frac{1}{8} \cdot \frac{1}{{x}^{2} \cdot {\left(\sqrt{\frac{1}{x}}\right)}^{3}}}{x} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 5.1%

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

Alternative 3: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\\ t_2 := \sqrt{\frac{t - -1}{t}} - 1\\ t_3 := \sqrt{y - -1}\\ \mathbf{if}\;x \leq 104000000:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \sqrt{t}, \left(\left(\sqrt{x - -1} - \sqrt{x}\right) - \frac{\sqrt{y} \cdot \sqrt{y} - t\_3 \cdot t\_3}{\sqrt{y} + t\_3}\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \sqrt{t}, \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - \left(\sqrt{y} - t\_3\right)\right) - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ -1.0 (+ (sqrt (- z -1.0)) (sqrt z))))
        (t_2 (- (sqrt (/ (- t -1.0) t)) 1.0))
        (t_3 (sqrt (- y -1.0))))
   (if (<= x 104000000.0)
     (fma
      t_2
      (sqrt t)
      (-
       (-
        (- (sqrt (- x -1.0)) (sqrt x))
        (/ (- (* (sqrt y) (sqrt y)) (* t_3 t_3)) (+ (sqrt y) t_3)))
       t_1))
     (fma
      t_2
      (sqrt t)
      (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt y) t_3)) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -1.0 / (sqrt((z - -1.0)) + sqrt(z));
	double t_2 = sqrt(((t - -1.0) / t)) - 1.0;
	double t_3 = sqrt((y - -1.0));
	double tmp;
	if (x <= 104000000.0) {
		tmp = fma(t_2, sqrt(t), (((sqrt((x - -1.0)) - sqrt(x)) - (((sqrt(y) * sqrt(y)) - (t_3 * t_3)) / (sqrt(y) + t_3))) - t_1));
	} else {
		tmp = fma(t_2, sqrt(t), (((0.5 / (x * sqrt((1.0 / x)))) - (sqrt(y) - t_3)) - t_1));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(-1.0 / Float64(sqrt(Float64(z - -1.0)) + sqrt(z)))
	t_2 = Float64(sqrt(Float64(Float64(t - -1.0) / t)) - 1.0)
	t_3 = sqrt(Float64(y - -1.0))
	tmp = 0.0
	if (x <= 104000000.0)
		tmp = fma(t_2, sqrt(t), Float64(Float64(Float64(sqrt(Float64(x - -1.0)) - sqrt(x)) - Float64(Float64(Float64(sqrt(y) * sqrt(y)) - Float64(t_3 * t_3)) / Float64(sqrt(y) + t_3))) - t_1));
	else
		tmp = fma(t_2, sqrt(t), Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - Float64(sqrt(y) - t_3)) - t_1));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-1.0 / N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(t - -1.0), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 104000000.0], N[(t$95$2 * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04e8

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-+.f64 93.7

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

   9. Applied rewrites 93.7%

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

   if 1.04e8 < x

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{t - -1}{t}} - 1, \sqrt{t}, \left(\color{blue}{\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{y} - \sqrt{y - -1}\right)\right) - \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   10. Applied rewrites 10.2%

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

Alternative 4: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} - \sqrt{y - -1}\\ t_2 := \sqrt{\frac{t - -1}{t}} - 1\\ t_3 := \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\\ \mathbf{if}\;x \leq 104000000:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \sqrt{t}, \left(\left(\sqrt{x - -1} - \sqrt{x}\right) - t\_1\right) - t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \sqrt{t}, \left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} - t\_1\right) - t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt y) (sqrt (- y -1.0))))
        (t_2 (- (sqrt (/ (- t -1.0) t)) 1.0))
        (t_3 (/ -1.0 (+ (sqrt (- z -1.0)) (sqrt z)))))
   (if (<= x 104000000.0)
     (fma t_2 (sqrt t) (- (- (- (sqrt (- x -1.0)) (sqrt x)) t_1) t_3))
     (fma t_2 (sqrt t) (- (- (/ 0.5 (* x (sqrt (/ 1.0 x)))) t_1) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) - sqrt((y - -1.0));
	double t_2 = sqrt(((t - -1.0) / t)) - 1.0;
	double t_3 = -1.0 / (sqrt((z - -1.0)) + sqrt(z));
	double tmp;
	if (x <= 104000000.0) {
		tmp = fma(t_2, sqrt(t), (((sqrt((x - -1.0)) - sqrt(x)) - t_1) - t_3));
	} else {
		tmp = fma(t_2, sqrt(t), (((0.5 / (x * sqrt((1.0 / x)))) - t_1) - t_3));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) - sqrt(Float64(y - -1.0)))
	t_2 = Float64(sqrt(Float64(Float64(t - -1.0) / t)) - 1.0)
	t_3 = Float64(-1.0 / Float64(sqrt(Float64(z - -1.0)) + sqrt(z)))
	tmp = 0.0
	if (x <= 104000000.0)
		tmp = fma(t_2, sqrt(t), Float64(Float64(Float64(sqrt(Float64(x - -1.0)) - sqrt(x)) - t_1) - t_3));
	else
		tmp = fma(t_2, sqrt(t), Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) - t_1) - t_3));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(t - -1.0), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 104000000.0], N[(t$95$2 * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04e8

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

   if 1.04e8 < x

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{t - -1}{t}} - 1, \sqrt{t}, \left(\color{blue}{\frac{\frac{1}{2}}{x \cdot \sqrt{\frac{1}{x}}}} - \left(\sqrt{y} - \sqrt{y - -1}\right)\right) - \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   10. Applied rewrites 10.2%

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

Alternative 5: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 135000000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{t - -1}{t}} - 1, \sqrt{t}, \left(\left(\sqrt{x - -1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y - -1}\right)\right) - \frac{-1}{\sqrt{z - -1} + \sqrt{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 135000000000.0)
   (fma
    (- (sqrt (/ (- t -1.0) t)) 1.0)
    (sqrt t)
    (-
     (- (- (sqrt (- x -1.0)) (sqrt x)) (- (sqrt y) (sqrt (- y -1.0))))
     (/ -1.0 (+ (sqrt (- z -1.0)) (sqrt z)))))
   (+
    (+ (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt (+ z 1.0)) (sqrt z)))
    (- (sqrt (+ t 1.0)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 135000000000.0) {
		tmp = fma((sqrt(((t - -1.0) / t)) - 1.0), sqrt(t), (((sqrt((x - -1.0)) - sqrt(x)) - (sqrt(y) - sqrt((y - -1.0)))) - (-1.0 / (sqrt((z - -1.0)) + sqrt(z)))));
	} else {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 135000000000.0)
		tmp = fma(Float64(sqrt(Float64(Float64(t - -1.0) / t)) - 1.0), sqrt(t), Float64(Float64(Float64(sqrt(Float64(x - -1.0)) - sqrt(x)) - Float64(sqrt(y) - sqrt(Float64(y - -1.0)))) - Float64(-1.0 / Float64(sqrt(Float64(z - -1.0)) + sqrt(z)))));
	else
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 135000000000.0], N[(N[(N[Sqrt[N[(N[(t - -1.0), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[t], $MachinePrecision] + N[(N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $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]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e11

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

   if 1.35e11 < x

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

Alternative 6: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z - -1}\\ t_2 := \sqrt{y - -1}\\ t_3 := \sqrt{z + 1} - \sqrt{z}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_3\right) + t\_4\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + t\_3\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 3:\\ \;\;\;\;\left(\sqrt{x - -1} - \left(\left(\sqrt{y} - \left(t\_1 - \sqrt{z}\right)\right) - t\_2\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 - \left(\sqrt{x} - 1\right)\right) - \left(\sqrt{y} - t\_1\right)\right) - \left(\sqrt{z} - \left(\sqrt{t - -1} - \sqrt{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (- z -1.0)))
        (t_2 (sqrt (- y -1.0)))
        (t_3 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_3)
          t_4)))
   (if (<= t_5 2e-6)
     (+ (+ (/ 0.5 (* x (sqrt (/ 1.0 x)))) t_3) t_4)
     (if (<= t_5 3.0)
       (- (- (sqrt (- x -1.0)) (- (- (sqrt y) (- t_1 (sqrt z))) t_2)) (sqrt x))
       (-
        (- (- t_2 (- (sqrt x) 1.0)) (- (sqrt y) t_1))
        (- (sqrt z) (- (sqrt (- t -1.0)) (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z - -1.0));
	double t_2 = sqrt((y - -1.0));
	double t_3 = sqrt((z + 1.0)) - sqrt(z);
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_3) + t_4;
	double tmp;
	if (t_5 <= 2e-6) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_3) + t_4;
	} else if (t_5 <= 3.0) {
		tmp = (sqrt((x - -1.0)) - ((sqrt(y) - (t_1 - sqrt(z))) - t_2)) - sqrt(x);
	} else {
		tmp = ((t_2 - (sqrt(x) - 1.0)) - (sqrt(y) - t_1)) - (sqrt(z) - (sqrt((t - -1.0)) - sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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((z - (-1.0d0)))
    t_2 = sqrt((y - (-1.0d0)))
    t_3 = sqrt((z + 1.0d0)) - sqrt(z)
    t_4 = sqrt((t + 1.0d0)) - sqrt(t)
    t_5 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_3) + t_4
    if (t_5 <= 2d-6) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) + t_3) + t_4
    else if (t_5 <= 3.0d0) then
        tmp = (sqrt((x - (-1.0d0))) - ((sqrt(y) - (t_1 - sqrt(z))) - t_2)) - sqrt(x)
    else
        tmp = ((t_2 - (sqrt(x) - 1.0d0)) - (sqrt(y) - t_1)) - (sqrt(z) - (sqrt((t - (-1.0d0))) - sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z - -1.0));
	double t_2 = Math.sqrt((y - -1.0));
	double t_3 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_4 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_5 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_3) + t_4;
	double tmp;
	if (t_5 <= 2e-6) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) + t_3) + t_4;
	} else if (t_5 <= 3.0) {
		tmp = (Math.sqrt((x - -1.0)) - ((Math.sqrt(y) - (t_1 - Math.sqrt(z))) - t_2)) - Math.sqrt(x);
	} else {
		tmp = ((t_2 - (Math.sqrt(x) - 1.0)) - (Math.sqrt(y) - t_1)) - (Math.sqrt(z) - (Math.sqrt((t - -1.0)) - Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z - -1.0))
	t_2 = math.sqrt((y - -1.0))
	t_3 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_4 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_5 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_3) + t_4
	tmp = 0
	if t_5 <= 2e-6:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) + t_3) + t_4
	elif t_5 <= 3.0:
		tmp = (math.sqrt((x - -1.0)) - ((math.sqrt(y) - (t_1 - math.sqrt(z))) - t_2)) - math.sqrt(x)
	else:
		tmp = ((t_2 - (math.sqrt(x) - 1.0)) - (math.sqrt(y) - t_1)) - (math.sqrt(z) - (math.sqrt((t - -1.0)) - math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z - -1.0))
	t_2 = sqrt(Float64(y - -1.0))
	t_3 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_3) + t_4)
	tmp = 0.0
	if (t_5 <= 2e-6)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) + t_3) + t_4);
	elseif (t_5 <= 3.0)
		tmp = Float64(Float64(sqrt(Float64(x - -1.0)) - Float64(Float64(sqrt(y) - Float64(t_1 - sqrt(z))) - t_2)) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(t_2 - Float64(sqrt(x) - 1.0)) - Float64(sqrt(y) - t_1)) - Float64(sqrt(z) - Float64(sqrt(Float64(t - -1.0)) - sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z - -1.0));
	t_2 = sqrt((y - -1.0));
	t_3 = sqrt((z + 1.0)) - sqrt(z);
	t_4 = sqrt((t + 1.0)) - sqrt(t);
	t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_3) + t_4;
	tmp = 0.0;
	if (t_5 <= 2e-6)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_3) + t_4;
	elseif (t_5 <= 3.0)
		tmp = (sqrt((x - -1.0)) - ((sqrt(y) - (t_1 - sqrt(z))) - t_2)) - sqrt(x);
	else
		tmp = ((t_2 - (sqrt(x) - 1.0)) - (sqrt(y) - t_1)) - (sqrt(z) - (sqrt((t - -1.0)) - sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 2e-6], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 3.0], N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < 1.99999999999999991e-6

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

   if 1.99999999999999991e-6 < (+.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))) < 3

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   8. Applied rewrites 86.7%

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

   if 3 < (+.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 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   3. Applied rewrites 39.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 39.8%

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

Alternative 7: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 135000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= x 135000000000.0)
     (+
      (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) t_1)
      t_2)
     (+ (+ (/ 0.5 (* x (sqrt (/ 1.0 x)))) t_1) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (x <= 135000000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	} else {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_1) + t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    if (x <= 135000000000.0d0) then
        tmp = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_2
    else
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) + t_1) + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (x <= 135000000000.0) {
		tmp = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_2;
	} else {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) + t_1) + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if x <= 135000000000.0:
		tmp = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_2
	else:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) + t_1) + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (x <= 135000000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_2);
	else
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) + t_1) + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (x <= 135000000000.0)
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	else
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_1) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 135000000000.0], 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] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e11

   1. Initial program 92.1%

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

   if 1.35e11 < x

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

Alternative 8: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\left(\left(\sqrt{y - -1} - \sqrt{y}\right) - \left(\sqrt{x} - \mathsf{fma}\left(0.5, x, 1\right)\right)\right) - \left(\left(\sqrt{z} - \sqrt{z - -1}\right) - \left(\sqrt{t - -1} - \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.4)
   (-
    (- (- (sqrt (- y -1.0)) (sqrt y)) (- (sqrt x) (fma 0.5 x 1.0)))
    (- (- (sqrt z) (sqrt (- z -1.0))) (- (sqrt (- t -1.0)) (sqrt t))))
   (+
    (+ (/ 0.5 (* x (sqrt (/ 1.0 x)))) (- (sqrt (+ z 1.0)) (sqrt z)))
    (- (sqrt (+ t 1.0)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.4) {
		tmp = ((sqrt((y - -1.0)) - sqrt(y)) - (sqrt(x) - fma(0.5, x, 1.0))) - ((sqrt(z) - sqrt((z - -1.0))) - (sqrt((t - -1.0)) - sqrt(t)));
	} else {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(Float64(Float64(sqrt(Float64(y - -1.0)) - sqrt(y)) - Float64(sqrt(x) - fma(0.5, x, 1.0))) - Float64(Float64(sqrt(z) - sqrt(Float64(z - -1.0))) - Float64(sqrt(Float64(t - -1.0)) - sqrt(t))));
	else
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 3.4], N[(N[(N[(N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - N[(0.5 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(t - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $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]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

   6. Applied rewrites 91.4%

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

   if 3.39999999999999991 < x

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

Alternative 9: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    if (x <= 1.0d0) then
        tmp = (((1.0d0 - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_2
    else
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) + t_1) + t_2
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.8%

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

   if 1 < x

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

Alternative 10: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{0.5}{x \cdot \sqrt{\frac{1}{x}}} + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x - -1} - \left(\left(\sqrt{y} - \left(\sqrt{z - -1} - \sqrt{z}\right)\right) - \sqrt{y - -1}\right)\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z))))
   (if (<=
        (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) t_1)
        2e-6)
     (+ (+ (/ 0.5 (* x (sqrt (/ 1.0 x)))) t_1) (- (sqrt (+ t 1.0)) (sqrt t)))
     (-
      (-
       (sqrt (- x -1.0))
       (- (- (sqrt y) (- (sqrt (- z -1.0)) (sqrt z))) (sqrt (- y -1.0))))
      (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double tmp;
	if ((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) <= 2e-6) {
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (sqrt((x - -1.0)) - ((sqrt(y) - (sqrt((z - -1.0)) - sqrt(z))) - sqrt((y - -1.0)))) - sqrt(x);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    if ((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) <= 2d-6) then
        tmp = ((0.5d0 / (x * sqrt((1.0d0 / x)))) + t_1) + (sqrt((t + 1.0d0)) - sqrt(t))
    else
        tmp = (sqrt((x - (-1.0d0))) - ((sqrt(y) - (sqrt((z - (-1.0d0))) - sqrt(z))) - sqrt((y - (-1.0d0))))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double tmp;
	if ((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) <= 2e-6) {
		tmp = ((0.5 / (x * Math.sqrt((1.0 / x)))) + t_1) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else {
		tmp = (Math.sqrt((x - -1.0)) - ((Math.sqrt(y) - (Math.sqrt((z - -1.0)) - Math.sqrt(z))) - Math.sqrt((y - -1.0)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	tmp = 0
	if (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) <= 2e-6:
		tmp = ((0.5 / (x * math.sqrt((1.0 / x)))) + t_1) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	else:
		tmp = (math.sqrt((x - -1.0)) - ((math.sqrt(y) - (math.sqrt((z - -1.0)) - math.sqrt(z))) - math.sqrt((y - -1.0)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	tmp = 0.0
	if (Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) <= 2e-6)
		tmp = Float64(Float64(Float64(0.5 / Float64(x * sqrt(Float64(1.0 / x)))) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(sqrt(Float64(x - -1.0)) - Float64(Float64(sqrt(y) - Float64(sqrt(Float64(z - -1.0)) - sqrt(z))) - sqrt(Float64(y - -1.0)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	tmp = 0.0;
	if ((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) <= 2e-6)
		tmp = ((0.5 / (x * sqrt((1.0 / x)))) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	else
		tmp = (sqrt((x - -1.0)) - ((sqrt(y) - (sqrt((z - -1.0)) - sqrt(z))) - sqrt((y - -1.0)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] + t$95$1), $MachinePrecision], 2e-6], N[(N[(N[(0.5 / N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] - N[(N[Sqrt[N[(z - -1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(y - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.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))) < 1.99999999999999991e-6

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 10.2

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

   7. Applied rewrites 10.2%

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

   if 1.99999999999999991e-6 < (+.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)))

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   8. Applied rewrites 86.7%

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

Alternative 11: 86.7% accurate, 1.4× speedup

Math

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

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 -1.0))
   (- (- (sqrt y) (- (sqrt (- z -1.0)) (sqrt z))) (sqrt (- y -1.0))))
  (sqrt x)))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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(y) - (sqrt((z - (-1.0d0))) - sqrt(z))) - sqrt((y - (-1.0d0))))) - sqrt(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Applied rewrites 92.1%

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

4. Taylor expanded in t around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 33.6

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

6. Applied rewrites 33.6%

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

8. Applied rewrites 86.7%

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

Alternative 12: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := -\sqrt{x}\\ t_3 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 \leq 1.00000005:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_2 + \left(\left(t\_4 + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_4 + \left(1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt x)))
        (t_3
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_1))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= t_3 1.00000005)
     (+ (+ (- 1.0 (sqrt x)) t_1) (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_3 2.0)
       (+ t_2 (- (+ t_4 (sqrt (+ 1.0 y))) (sqrt y)))
       (+ t_2 (- (+ t_4 (+ 1.0 (sqrt (+ 1.0 z)))) (+ (sqrt y) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = -sqrt(x);
	double t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 1.00000005) {
		tmp = ((1.0 - sqrt(x)) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_3 <= 2.0) {
		tmp = t_2 + ((t_4 + sqrt((1.0 + y))) - sqrt(y));
	} else {
		tmp = t_2 + ((t_4 + (1.0 + sqrt((1.0 + z)))) - (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = -sqrt(x)
    t_3 = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1
    t_4 = sqrt((1.0d0 + x))
    if (t_3 <= 1.00000005d0) then
        tmp = ((1.0d0 - sqrt(x)) + t_1) + (sqrt((t + 1.0d0)) - sqrt(t))
    else if (t_3 <= 2.0d0) then
        tmp = t_2 + ((t_4 + sqrt((1.0d0 + y))) - sqrt(y))
    else
        tmp = t_2 + ((t_4 + (1.0d0 + sqrt((1.0d0 + z)))) - (sqrt(y) + sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = -Math.sqrt(x);
	double t_3 = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1;
	double t_4 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 1.00000005) {
		tmp = ((1.0 - Math.sqrt(x)) + t_1) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else if (t_3 <= 2.0) {
		tmp = t_2 + ((t_4 + Math.sqrt((1.0 + y))) - Math.sqrt(y));
	} else {
		tmp = t_2 + ((t_4 + (1.0 + Math.sqrt((1.0 + z)))) - (Math.sqrt(y) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = -math.sqrt(x)
	t_3 = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_3 <= 1.00000005:
		tmp = ((1.0 - math.sqrt(x)) + t_1) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	elif t_3 <= 2.0:
		tmp = t_2 + ((t_4 + math.sqrt((1.0 + y))) - math.sqrt(y))
	else:
		tmp = t_2 + ((t_4 + (1.0 + math.sqrt((1.0 + z)))) - (math.sqrt(y) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(-sqrt(x))
	t_3 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1)
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_3 <= 1.00000005)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_3 <= 2.0)
		tmp = Float64(t_2 + Float64(Float64(t_4 + sqrt(Float64(1.0 + y))) - sqrt(y)));
	else
		tmp = Float64(t_2 + Float64(Float64(t_4 + Float64(1.0 + sqrt(Float64(1.0 + z)))) - Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = -sqrt(x);
	t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_3 <= 1.00000005)
		tmp = ((1.0 - sqrt(x)) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	elseif (t_3 <= 2.0)
		tmp = t_2 + ((t_4 + sqrt((1.0 + y))) - sqrt(y));
	else
		tmp = t_2 + ((t_4 + (1.0 + sqrt((1.0 + z)))) - (sqrt(y) + 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[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[x], $MachinePrecision])}, Block[{t$95$3 = 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] + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1.00000005], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$2 + N[(N[(t$95$4 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$4 + N[(1.0 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 1.00000004999999992

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   if 1.00000004999999992 < (+.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))) < 2

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 47.3

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

   9. Applied rewrites 47.3%

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

   if 2 < (+.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)))

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 33.0

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

   9. Applied rewrites 33.0%

      \[\leadsto \left(-\sqrt{x}\right) + \left(\left(\sqrt{1 + x} + \left(1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{1 + y}\\ t_3 := -\sqrt{x}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_4\\ t_6 := \sqrt{1 + x}\\ \mathbf{if}\;t\_5 \leq 1.00000005:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + t\_1\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.5:\\ \;\;\;\;t\_3 + \left(\left(t\_6 + t\_2\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(t\_6 + \left(1 + t\_2\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (- (sqrt x)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_1)
          t_4))
        (t_6 (sqrt (+ 1.0 x))))
   (if (<= t_5 1.00000005)
     (+ (+ (- 1.0 (sqrt x)) t_1) t_4)
     (if (<= t_5 2.5)
       (+ t_3 (- (+ t_6 t_2) (sqrt y)))
       (+ t_3 (- (+ t_6 (+ 1.0 t_2)) (+ (sqrt y) (sqrt z))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((1.0 + y));
	double t_3 = -sqrt(x);
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_4;
	double t_6 = sqrt((1.0 + x));
	double tmp;
	if (t_5 <= 1.00000005) {
		tmp = ((1.0 - sqrt(x)) + t_1) + t_4;
	} else if (t_5 <= 2.5) {
		tmp = t_3 + ((t_6 + t_2) - sqrt(y));
	} else {
		tmp = t_3 + ((t_6 + (1.0 + t_2)) - (sqrt(y) + sqrt(z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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) :: t_6
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((1.0d0 + y))
    t_3 = -sqrt(x)
    t_4 = sqrt((t + 1.0d0)) - sqrt(t)
    t_5 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_4
    t_6 = sqrt((1.0d0 + x))
    if (t_5 <= 1.00000005d0) then
        tmp = ((1.0d0 - sqrt(x)) + t_1) + t_4
    else if (t_5 <= 2.5d0) then
        tmp = t_3 + ((t_6 + t_2) - sqrt(y))
    else
        tmp = t_3 + ((t_6 + (1.0d0 + t_2)) - (sqrt(y) + sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((1.0 + y));
	double t_3 = -Math.sqrt(x);
	double t_4 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_5 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_4;
	double t_6 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_5 <= 1.00000005) {
		tmp = ((1.0 - Math.sqrt(x)) + t_1) + t_4;
	} else if (t_5 <= 2.5) {
		tmp = t_3 + ((t_6 + t_2) - Math.sqrt(y));
	} else {
		tmp = t_3 + ((t_6 + (1.0 + t_2)) - (Math.sqrt(y) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((1.0 + y))
	t_3 = -math.sqrt(x)
	t_4 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_5 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_4
	t_6 = math.sqrt((1.0 + x))
	tmp = 0
	if t_5 <= 1.00000005:
		tmp = ((1.0 - math.sqrt(x)) + t_1) + t_4
	elif t_5 <= 2.5:
		tmp = t_3 + ((t_6 + t_2) - math.sqrt(y))
	else:
		tmp = t_3 + ((t_6 + (1.0 + t_2)) - (math.sqrt(y) + math.sqrt(z)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + y))
	t_3 = Float64(-sqrt(x))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_4)
	t_6 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_5 <= 1.00000005)
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + t_1) + t_4);
	elseif (t_5 <= 2.5)
		tmp = Float64(t_3 + Float64(Float64(t_6 + t_2) - sqrt(y)));
	else
		tmp = Float64(t_3 + Float64(Float64(t_6 + Float64(1.0 + t_2)) - Float64(sqrt(y) + sqrt(z))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((1.0 + y));
	t_3 = -sqrt(x);
	t_4 = sqrt((t + 1.0)) - sqrt(t);
	t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_4;
	t_6 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_5 <= 1.00000005)
		tmp = ((1.0 - sqrt(x)) + t_1) + t_4;
	elseif (t_5 <= 2.5)
		tmp = t_3 + ((t_6 + t_2) - sqrt(y));
	else
		tmp = t_3 + ((t_6 + (1.0 + t_2)) - (sqrt(y) + 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[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[x], $MachinePrecision])}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 1.00000005], N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.5], N[(t$95$3 + N[(N[(t$95$6 + t$95$2), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(t$95$6 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 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))) < 1.00000004999999992

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

   if 1.00000004999999992 < (+.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))) < 2.5

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 47.3

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

   9. Applied rewrites 47.3%

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

   if 2.5 < (+.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 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 25.4

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

   9. Applied rewrites 25.4%

      \[\leadsto \left(-\sqrt{x}\right) + \left(\left(\sqrt{1 + x} + \left(1 + \sqrt{1 + y}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 64.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+33}:\\ \;\;\;\;\left(-\sqrt{x}\right) + \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.15e+33)
   (+ (- (sqrt x)) (- (+ (sqrt (+ 1.0 x)) (sqrt (+ 1.0 y))) (sqrt y)))
   (+
    (+ (- 1.0 (sqrt x)) (- (sqrt (+ z 1.0)) (sqrt z)))
    (- (sqrt (+ t 1.0)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.15e+33) {
		tmp = -sqrt(x) + ((sqrt((1.0 + x)) + sqrt((1.0 + y))) - sqrt(y));
	} else {
		tmp = ((1.0 - sqrt(x)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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 <= 2.15d+33) then
        tmp = -sqrt(x) + ((sqrt((1.0d0 + x)) + sqrt((1.0d0 + y))) - sqrt(y))
    else
        tmp = ((1.0d0 - sqrt(x)) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.15e+33) {
		tmp = -Math.sqrt(x) + ((Math.sqrt((1.0 + x)) + Math.sqrt((1.0 + y))) - Math.sqrt(y));
	} else {
		tmp = ((1.0 - Math.sqrt(x)) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 2.15e+33:
		tmp = -math.sqrt(x) + ((math.sqrt((1.0 + x)) + math.sqrt((1.0 + y))) - math.sqrt(y))
	else:
		tmp = ((1.0 - math.sqrt(x)) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.15e+33)
		tmp = Float64(Float64(-sqrt(x)) + Float64(Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 + y))) - sqrt(y)));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.15e+33)
		tmp = -sqrt(x) + ((sqrt((1.0 + x)) + sqrt((1.0 + y))) - sqrt(y));
	else
		tmp = ((1.0 - sqrt(x)) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 2.15e+33], N[((-N[Sqrt[x], $MachinePrecision]) + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[x], $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]]

Derivation

1. Split input into 2 regimes

2. if y < 2.15000000000000014e33

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 47.3

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

   9. Applied rewrites 47.3%

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

   if 2.15000000000000014e33 < y

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 36.5

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 35.3

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

   7. Applied rewrites 35.3%

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

Alternative 15: 64.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{{\left(\sqrt{t}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{x}\right) + \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (<= (- (sqrt (+ y 1.0)) (sqrt y)) 2e-8)
   (/ (pow (sqrt t) 2.0) t)
   (+ (- (sqrt x)) (- (+ (sqrt (+ 1.0 x)) (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 ((sqrt((y + 1.0)) - sqrt(y)) <= 2e-8) {
		tmp = pow(sqrt(t), 2.0) / t;
	} else {
		tmp = -sqrt(x) + ((sqrt((1.0 + x)) + 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((sqrt((y + 1.0d0)) - sqrt(y)) <= 2d-8) then
        tmp = (sqrt(t) ** 2.0d0) / t
    else
        tmp = -sqrt(x) + ((sqrt((1.0d0 + x)) + 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 ((Math.sqrt((y + 1.0)) - Math.sqrt(y)) <= 2e-8) {
		tmp = Math.pow(Math.sqrt(t), 2.0) / t;
	} else {
		tmp = -Math.sqrt(x) + ((Math.sqrt((1.0 + x)) + 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 (math.sqrt((y + 1.0)) - math.sqrt(y)) <= 2e-8:
		tmp = math.pow(math.sqrt(t), 2.0) / t
	else:
		tmp = -math.sqrt(x) + ((math.sqrt((1.0 + x)) + 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 (Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) <= 2e-8)
		tmp = Float64((sqrt(t) ^ 2.0) / t);
	else
		tmp = Float64(Float64(-sqrt(x)) + Float64(Float64(sqrt(Float64(1.0 + x)) + 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 ((sqrt((y + 1.0)) - sqrt(y)) <= 2e-8)
		tmp = (sqrt(t) ^ 2.0) / t;
	else
		tmp = -sqrt(x) + ((sqrt((1.0 + x)) + 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[N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Power[N[Sqrt[t], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], N[((-N[Sqrt[x], $MachinePrecision]) + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 2e-8

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-+.f64 65.9

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 65.9

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

   3. Applied rewrites 65.9%

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{t}\right)}^{2}}{t}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{\color{blue}{t}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{t} \]

      3. lower-sqrt.f64 34.3

         \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{t} \]

   10. Applied rewrites 34.3%

       \[\leadsto \color{blue}{\frac{{\left(\sqrt{t}\right)}^{2}}{t}} \]

   if 2e-8 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 92.1%

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

   3. Applied rewrites 92.1%

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

   4. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 47.3

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

   9. Applied rewrites 47.3%

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

Alternative 16: 34.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{{\left(\sqrt{t}\right)}^{2}}{t} \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return pow(sqrt(t), 2.0) / t;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt(t) ** 2.0d0) / t
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.pow(Math.sqrt(t), 2.0) / t;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.pow(math.sqrt(t), 2.0) / t

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64((sqrt(t) ^ 2.0) / t)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt(t) ^ 2.0) / 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[(N[Power[N[Sqrt[t], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-+.f64 N/A

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

   11. add-flip N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-unsound-*.f64 N/A

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

   15. lower-unsound-+.f64 65.9

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

   16. lift-+.f64 N/A

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

   17. add-flip N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval 65.9

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

3. Applied rewrites 65.9%

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

5. Applied rewrites 92.7%

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

7. Applied rewrites 93.5%

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

8. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\left(\sqrt{t}\right)}^{2}}{t}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{\color{blue}{t}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{t} \]

   3. lower-sqrt.f64 34.3

      \[\leadsto \frac{{\left(\sqrt{t}\right)}^{2}}{t} \]

10. Applied rewrites 34.3%

    \[\leadsto \color{blue}{\frac{{\left(\sqrt{t}\right)}^{2}}{t}} \]
11. Add Preprocessing

Alternative 17: 7.7% accurate, 7.8× speedup

Math

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

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 z)))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.5d0 * sqrt(z)
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(z);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-+.f64 N/A

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

   11. add-flip N/A

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

   12. lower--.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-unsound-*.f64 N/A

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

   15. lower-unsound-+.f64 65.9

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

   16. lift-+.f64 N/A

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

   17. add-flip N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval 65.9

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

3. Applied rewrites 65.9%

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

4. Taylor expanded in z around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(z \cdot \sqrt{\frac{1}{z}}\right) \]

   4. lower-/.f64 7.7

      \[\leadsto 0.5 \cdot \left(z \cdot \sqrt{\frac{1}{z}}\right) \]

6. Applied rewrites 7.7%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{z}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{z} \]

   2. lower-sqrt.f64 7.7

      \[\leadsto 0.5 \cdot \sqrt{z} \]

9. Applied rewrites 7.7%

   \[\leadsto 0.5 \cdot \color{blue}{\sqrt{z}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(FPCore (x y z t)
  :name "Main:z from "
  :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))))