Main:z from

Percentage Accurate: 91.9% → 97.2%

Time: 13.5s

Alternatives: 19

Speedup: 0.9×

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: 91.9% 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: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ \mathbf{if}\;x \leq 0.445:\\ \;\;\;\;\left(\left(\left(1 + t\_2\right) - \sqrt{x}\right) + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, t\_2\right) + t\_1\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
   (if (<= x 0.445)
     (+ (+ (- (+ 1.0 t_2) (sqrt x)) t_1) (- (sqrt (+ t 1.0)) (sqrt t)))
     (+ (+ (fma 0.5 (/ 1.0 (sqrt x)) t_2) t_1) (- (sqrt t) (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)) - sqrt(z);
	double t_2 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double tmp;
	if (x <= 0.445) {
		tmp = (((1.0 + t_2) - sqrt(x)) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (fma(0.5, (1.0 / sqrt(x)), t_2) + t_1) + (sqrt(t) - sqrt(t));
	}
	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(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))
	tmp = 0.0
	if (x <= 0.445)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_2) - sqrt(x)) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), t_2) + t_1) + Float64(sqrt(t) - 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[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.445], N[(N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.445000000000000007

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   if 0.445000000000000007 < x

   1. Initial program 13.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 13.9%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-div N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{\sqrt{1}}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 19.3

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

   6. Applied rewrites 19.3%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in x around inf

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

       1. lower-fma.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lift-sqrt.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 90.8

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

   12. Applied rewrites 90.8%

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

Alternative 2: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;\left(\left(\left(1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{\sqrt{x}} + \frac{\mathsf{fma}\left(-0.125, \frac{1}{\sqrt{y}}, 0.5 \cdot \sqrt{y}\right)}{y}\right) + t\_1\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z))))
   (if (<= x 2.8)
     (+
      (+ (- (+ 1.0 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x)) t_1)
      (- (sqrt (+ t 1.0)) (sqrt t)))
     (+
      (+
       (+
        (* 0.5 (/ 1.0 (sqrt x)))
        (/ (fma -0.125 (/ 1.0 (sqrt y)) (* 0.5 (sqrt y))) y))
       t_1)
      (- (sqrt t) (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)) - sqrt(z);
	double tmp;
	if (x <= 2.8) {
		tmp = (((1.0 + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (((0.5 * (1.0 / sqrt(x))) + (fma(-0.125, (1.0 / sqrt(y)), (0.5 * sqrt(y))) / y)) + t_1) + (sqrt(t) - sqrt(t));
	}
	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 (x <= 2.8)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + Float64(fma(-0.125, Float64(1.0 / sqrt(y)), Float64(0.5 * sqrt(y))) / y)) + t_1) + Float64(sqrt(t) - 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[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.8], N[(N[(N[(N[(1.0 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if 2.7999999999999998 < x

   1. Initial program 13.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 13.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-div N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{\sqrt{1}}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 19.0

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

   6. Applied rewrites 19.0%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 19.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t} - \sqrt{t}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t} - \sqrt{t}\right) \]

       2. sqrt-div N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}} + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t} - \sqrt{t}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{\sqrt{\color{blue}{x}}} + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t} - \sqrt{t}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\sqrt{x}}} + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t} - \sqrt{t}\right) \]

       5. lift-sqrt.f64 91.1

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

   12. Applied rewrites 91.1%

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

Alternative 3: 96.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

   if 1 < x

   1. Initial program 13.6%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 16.7

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

   4. Applied rewrites 16.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 4: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_2\right) + t\_3\\ t_5 := 0.5 \cdot \frac{1}{\sqrt{z}}\\ \mathbf{if}\;t\_4 \leq 1.0001:\\ \;\;\;\;\left(t\_1 + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, t\_5\right)\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.0001:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{1 + y} + t\_5\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_2\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_2)
          t_3))
        (t_5 (* 0.5 (/ 1.0 (sqrt z)))))
   (if (<= t_4 1.0001)
     (- (+ t_1 (fma 0.5 (/ 1.0 (sqrt y)) t_5)) (sqrt x))
     (if (<= t_4 2.0001)
       (- (+ t_1 (+ (sqrt (+ 1.0 y)) t_5)) (+ (sqrt x) (sqrt y)))
       (+ (+ (+ (- (fma 0.5 x 1.0) (sqrt x)) (- 1.0 (sqrt y))) t_2) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((z + 1.0)) - sqrt(z);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2) + t_3;
	double t_5 = 0.5 * (1.0 / sqrt(z));
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = (t_1 + fma(0.5, (1.0 / sqrt(y)), t_5)) - sqrt(x);
	} else if (t_4 <= 2.0001) {
		tmp = (t_1 + (sqrt((1.0 + y)) + t_5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((fma(0.5, x, 1.0) - sqrt(x)) + (1.0 - sqrt(y))) + t_2) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_2) + t_3)
	t_5 = Float64(0.5 * Float64(1.0 / sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1.0001)
		tmp = Float64(Float64(t_1 + fma(0.5, Float64(1.0 / sqrt(y)), t_5)) - sqrt(x));
	elseif (t_4 <= 2.0001)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) + t_5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.5, x, 1.0) - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_2) + 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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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$2), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0001], N[(N[(t$95$1 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0001], N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $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))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 1.00009999999999999 < (+.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))) < 2.00010000000000021

   1. Initial program 96.6%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

   7. Applied rewrites 99.2%

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

   if 2.00010000000000021 < (+.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 98.5%

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

   2. 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. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.3%

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

Alternative 5: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1} - \sqrt{x}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;\left(\left(t\_2 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3 \leq 0.9998:\\ \;\;\;\;\left(\left(t\_2 + \frac{1}{1 + \sqrt{y}}\right) + t\_1\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

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 1.0)) (sqrt x)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= (+ (+ (+ t_2 (- (sqrt (+ y 1.0)) (sqrt y))) t_1) t_3) 0.9998)
     (+ (+ (+ t_2 (/ 1.0 (+ 1.0 (sqrt y)))) t_1) (- (sqrt t) (sqrt t)))
     (+
      (+ (- (+ 1.0 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt 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((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0)) - sqrt(x);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if ((((t_2 + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3) <= 0.9998) {
		tmp = ((t_2 + (1.0 / (1.0 + sqrt(y)))) + t_1) + (sqrt(t) - sqrt(t));
	} else {
		tmp = (((1.0 + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_1) + t_3;
	}
	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) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((x + 1.0d0)) - sqrt(x)
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    if ((((t_2 + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_3) <= 0.9998d0) then
        tmp = ((t_2 + (1.0d0 / (1.0d0 + sqrt(y)))) + t_1) + (sqrt(t) - sqrt(t))
    else
        tmp = (((1.0d0 + (1.0d0 / (sqrt(y) + sqrt((1.0d0 + y))))) - sqrt(x)) + t_1) + t_3
    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 + 1.0)) - Math.sqrt(x);
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if ((((t_2 + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_3) <= 0.9998) {
		tmp = ((t_2 + (1.0 / (1.0 + Math.sqrt(y)))) + t_1) + (Math.sqrt(t) - Math.sqrt(t));
	} else {
		tmp = (((1.0 + (1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y))))) - Math.sqrt(x)) + t_1) + t_3;
	}
	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 + 1.0)) - math.sqrt(x)
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if (((t_2 + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_3) <= 0.9998:
		tmp = ((t_2 + (1.0 / (1.0 + math.sqrt(y)))) + t_1) + (math.sqrt(t) - math.sqrt(t))
	else:
		tmp = (((1.0 + (1.0 / (math.sqrt(y) + math.sqrt((1.0 + y))))) - math.sqrt(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(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3) <= 0.9998)
		tmp = Float64(Float64(Float64(t_2 + Float64(1.0 / Float64(1.0 + sqrt(y)))) + t_1) + Float64(sqrt(t) - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_1) + t_3);
	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 + 1.0)) - sqrt(x);
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if ((((t_2 + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3) <= 0.9998)
		tmp = ((t_2 + (1.0 / (1.0 + sqrt(y)))) + t_1) + (sqrt(t) - sqrt(t));
	else
		tmp = (((1.0 + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_1) + t_3;
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision], 0.9998], N[(N[(N[(t$95$2 + N[(1.0 / N[(1.0 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 0.99980000000000002

   1. Initial program 20.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 20.3%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-div N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{\sqrt{1}}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{y}}, \frac{1}{2} \cdot \sqrt{y}\right)}{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 25.5

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

   6. Applied rewrites 25.5%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 25.5%

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

   10. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 23.9

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

   12. Applied rewrites 23.9%

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

   if 0.99980000000000002 < (+.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 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 6: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2 \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_3) t_1) t_2) 1.0001)
     (-
      (+ (sqrt (+ 1.0 x)) (fma 0.5 (/ 1.0 (sqrt y)) (* 0.5 (/ 1.0 (sqrt z)))))
      (sqrt x))
     (+ (+ (+ (- (fma 0.5 x 1.0) (sqrt x)) t_3) 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 t_3 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + t_3) + t_1) + t_2) <= 1.0001) {
		tmp = (sqrt((1.0 + x)) + fma(0.5, (1.0 / sqrt(y)), (0.5 * (1.0 / sqrt(z))))) - sqrt(x);
	} else {
		tmp = (((fma(0.5, x, 1.0) - sqrt(x)) + t_3) + 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))
	t_3 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_3) + t_1) + t_2) <= 1.0001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + fma(0.5, Float64(1.0 / sqrt(y)), Float64(0.5 * Float64(1.0 / sqrt(z))))) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.5, x, 1.0) - sqrt(x)) + t_3) + t_1) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], 1.0001], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 1.00009999999999999 < (+.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 97.5%

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

   2. 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. +-commutative N/A

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

      2. lower-fma.f64 97.5

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

   4. Applied rewrites 97.5%

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

Alternative 7: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2 \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_3) t_1) t_2) 1.0001)
     (-
      (+ (sqrt (+ 1.0 x)) (fma 0.5 (/ 1.0 (sqrt y)) (* 0.5 (/ 1.0 (sqrt z)))))
      (sqrt x))
     (+ (+ (+ (- 1.0 (sqrt x)) t_3) 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 t_3 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + t_3) + t_1) + t_2) <= 1.0001) {
		tmp = (sqrt((1.0 + x)) + fma(0.5, (1.0 / sqrt(y)), (0.5 * (1.0 / sqrt(z))))) - sqrt(x);
	} else {
		tmp = (((1.0 - sqrt(x)) + t_3) + 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))
	t_3 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_3) + t_1) + t_2) <= 1.0001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + fma(0.5, Float64(1.0 / sqrt(y)), Float64(0.5 * Float64(1.0 / sqrt(z))))) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_3) + t_1) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], 1.0001], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 1.00009999999999999 < (+.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 97.5%

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

   2. 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 97.4%

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

Alternative 8: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0.999999999999:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= (- (sqrt (+ y 1.0)) (sqrt y)) 0.999999999999)
     (+
      (- (+ (sqrt (+ 1.0 x)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x))
      t_1)
     (+
      (+
       (+ (- (fma 0.5 x 1.0) (sqrt x)) (- 1.0 (sqrt y)))
       (- (sqrt (+ z 1.0)) (sqrt z)))
      t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if ((sqrt((y + 1.0)) - sqrt(y)) <= 0.999999999999) {
		tmp = ((sqrt((1.0 + x)) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_1;
	} else {
		tmp = (((fma(0.5, x, 1.0) - sqrt(x)) + (1.0 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) <= 0.999999999999)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_1);
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.5, x, 1.0) - sqrt(x)) + Float64(1.0 - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + 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[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.999999999999], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.999999999999000022

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 83.0

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

   6. Applied rewrites 83.0%

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

   if 0.999999999999000022 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 97.8%

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

   2. 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. +-commutative N/A

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

      2. lower-fma.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 97.8%

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

Alternative 9: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_5 := 0.5 \cdot \frac{1}{\sqrt{z}}\\ \mathbf{if}\;t\_4 \leq 1.0001:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, t\_5\right)\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.002:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + y} + t\_5\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_2 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \sqrt{z}\\ \end{array} \end{array} \]

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)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_5 (* 0.5 (/ 1.0 (sqrt z)))))
   (if (<= t_4 1.0001)
     (- (+ t_2 (fma 0.5 (/ 1.0 (sqrt y)) t_5)) (sqrt x))
     (if (<= t_4 2.002)
       (- (+ t_2 (+ (sqrt (+ 1.0 y)) t_5)) (+ (sqrt x) (sqrt y)))
       (- (- (+ (+ t_2 t_1) t_3) (sqrt x)) (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_5 = 0.5 * (1.0 / sqrt(z));
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = (t_2 + fma(0.5, (1.0 / sqrt(y)), t_5)) - sqrt(x);
	} else if (t_4 <= 2.002) {
		tmp = (t_2 + (sqrt((1.0 + y)) + t_5)) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	t_5 = Float64(0.5 * Float64(1.0 / sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1.0001)
		tmp = Float64(Float64(t_2 + fma(0.5, Float64(1.0 / sqrt(y)), t_5)) - sqrt(x));
	elseif (t_4 <= 2.002)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + y)) + t_5)) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z));
	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]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0001], N[(N[(t$95$2 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.002], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $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))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 1.00009999999999999 < (+.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))) < 2.0019999999999998

   1. Initial program 96.5%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 15.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

   7. Applied rewrites 98.8%

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

   if 2.0019999999999998 < (+.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 98.6%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 80.8

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

   7. Applied rewrites 80.8%

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

Alternative 10: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_4 \leq 1.0001:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{y}}, 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\left(t\_2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_2 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \sqrt{z}\\ \end{array} \end{array} \]

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)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.0001)
     (- (+ t_2 (fma 0.5 (/ 1.0 (sqrt y)) (* 0.5 (/ 1.0 (sqrt z))))) (sqrt x))
     (if (<= t_4 2.0)
       (- (+ t_2 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y)))
       (- (- (+ (+ t_2 t_1) t_3) (sqrt x)) (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = (t_2 + fma(0.5, (1.0 / sqrt(y)), (0.5 * (1.0 / sqrt(z))))) - sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = (t_2 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_4 <= 1.0001)
		tmp = Float64(Float64(t_2 + fma(0.5, Float64(1.0 / sqrt(y)), Float64(0.5 * Float64(1.0 / sqrt(z))))) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z));
	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]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0001], N[(N[(t$95$2 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $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))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 81.3%

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

   if 1.00009999999999999 < (+.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

   1. Initial program 97.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 11.4%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if 2 < (+.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 97.2%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 80.6

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

   7. Applied rewrites 80.6%

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

Alternative 11: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_4 \leq 1.0001:\\ \;\;\;\;\left(t\_2 + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\left(t\_2 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_2 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \sqrt{z}\\ \end{array} \end{array} \]

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)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.0001)
     (- (+ t_2 (* 0.5 (/ 1.0 (sqrt y)))) (sqrt x))
     (if (<= t_4 2.0)
       (- (+ t_2 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y)))
       (- (- (+ (+ t_2 t_1) t_3) (sqrt x)) (sqrt z))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = (t_2 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = (t_2 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((t_2 + t_1) + t_3) - sqrt(x)) - 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((y + 1.0d0))
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((z + 1.0d0))
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    if (t_4 <= 1.0001d0) then
        tmp = (t_2 + (0.5d0 * (1.0d0 / sqrt(y)))) - sqrt(x)
    else if (t_4 <= 2.0d0) then
        tmp = (t_2 + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y))
    else
        tmp = (((t_2 + t_1) + t_3) - sqrt(x)) - 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((y + 1.0));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + (t_3 - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	double tmp;
	if (t_4 <= 1.0001) {
		tmp = (t_2 + (0.5 * (1.0 / Math.sqrt(y)))) - Math.sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = (t_2 + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((t_2 + t_1) + t_3) - Math.sqrt(x)) - 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((y + 1.0))
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((z + 1.0))
	t_4 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (t_1 - math.sqrt(y))) + (t_3 - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	tmp = 0
	if t_4 <= 1.0001:
		tmp = (t_2 + (0.5 * (1.0 / math.sqrt(y)))) - math.sqrt(x)
	elif t_4 <= 2.0:
		tmp = (t_2 + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((t_2 + t_1) + t_3) - math.sqrt(x)) - math.sqrt(z)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_4 <= 1.0001)
		tmp = Float64(Float64(t_2 + Float64(0.5 * Float64(1.0 / sqrt(y)))) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(t_2 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 + t_1) + t_3) - sqrt(x)) - 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((y + 1.0));
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((z + 1.0));
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	tmp = 0.0;
	if (t_4 <= 1.0001)
		tmp = (t_2 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x);
	elseif (t_4 <= 2.0)
		tmp = (t_2 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	else
		tmp = (((t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0001], N[(N[(t$95$2 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(t$95$2 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $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))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 81.1

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

   10. Applied rewrites 81.1%

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

   if 1.00009999999999999 < (+.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

   1. Initial program 97.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 11.4%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if 2 < (+.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 97.2%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 80.6

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

   7. Applied rewrites 80.6%

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

Alternative 12: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ t_3 := t\_1 + \sqrt{1 + y}\\ \mathbf{if}\;t\_2 \leq 1.0001:\\ \;\;\;\;\left(t\_1 + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\\ \mathbf{elif}\;t\_2 \leq 2.5:\\ \;\;\;\;t\_3 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_3\right) - \sqrt{x}\right) - \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t))))
        (t_3 (+ t_1 (sqrt (+ 1.0 y)))))
   (if (<= t_2 1.0001)
     (- (+ t_1 (* 0.5 (/ 1.0 (sqrt y)))) (sqrt x))
     (if (<= t_2 2.5)
       (- t_3 (+ (sqrt x) (sqrt y)))
       (- (- (+ 1.0 t_3) (sqrt x)) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double t_3 = t_1 + sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0001) {
		tmp = (t_1 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x);
	} else if (t_2 <= 2.5) {
		tmp = t_3 - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((1.0 + t_3) - sqrt(x)) - 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    t_3 = t_1 + sqrt((1.0d0 + y))
    if (t_2 <= 1.0001d0) then
        tmp = (t_1 + (0.5d0 * (1.0d0 / sqrt(y)))) - sqrt(x)
    else if (t_2 <= 2.5d0) then
        tmp = t_3 - (sqrt(x) + sqrt(y))
    else
        tmp = ((1.0d0 + t_3) - sqrt(x)) - sqrt(y)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = (((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));
	double t_3 = t_1 + Math.sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 1.0001) {
		tmp = (t_1 + (0.5 * (1.0 / Math.sqrt(y)))) - Math.sqrt(x);
	} else if (t_2 <= 2.5) {
		tmp = t_3 - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = ((1.0 + t_3) - Math.sqrt(x)) - Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = (((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))
	t_3 = t_1 + math.sqrt((1.0 + y))
	tmp = 0
	if t_2 <= 1.0001:
		tmp = (t_1 + (0.5 * (1.0 / math.sqrt(y)))) - math.sqrt(x)
	elif t_2 <= 2.5:
		tmp = t_3 - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = ((1.0 + t_3) - math.sqrt(x)) - math.sqrt(y)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = 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)))
	t_3 = Float64(t_1 + sqrt(Float64(1.0 + y)))
	tmp = 0.0
	if (t_2 <= 1.0001)
		tmp = Float64(Float64(t_1 + Float64(0.5 * Float64(1.0 / sqrt(y)))) - sqrt(x));
	elseif (t_2 <= 2.5)
		tmp = Float64(t_3 - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(1.0 + t_3) - sqrt(x)) - sqrt(y));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	t_3 = t_1 + sqrt((1.0 + y));
	tmp = 0.0;
	if (t_2 <= 1.0001)
		tmp = (t_1 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x);
	elseif (t_2 <= 2.5)
		tmp = t_3 - (sqrt(x) + sqrt(y));
	else
		tmp = ((1.0 + t_3) - sqrt(x)) - sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1.0001], N[(N[(t$95$1 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.5], N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $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))) < 1.00009999999999999

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 81.1

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

   10. Applied rewrites 81.1%

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

   if 1.00009999999999999 < (+.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 96.5%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 17.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 92.6

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

   7. Applied rewrites 92.6%

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

   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 98.7%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in y around inf

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

      1. lift-sqrt.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in y around inf

      \[\leadsto \sqrt{y} - \sqrt{y} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 3.1

         \[\leadsto \sqrt{y} - \sqrt{y} \]

   10. Applied rewrites 3.1%

       \[\leadsto \sqrt{y} - \sqrt{y} \]

   11. Taylor expanded in z around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lift-sqrt.f64 72.7

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

   13. Applied rewrites 72.7%

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

Alternative 13: 66.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 66000000:\\ \;\;\;\;\left(t\_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))))
   (if (<= y 66000000.0)
     (- (+ t_1 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y)))
     (- (+ t_1 (* 0.5 (/ 1.0 (sqrt y)))) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double tmp;
	if (y <= 66000000.0) {
		tmp = (t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (t_1 + (0.5 * (1.0 / sqrt(y)))) - 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((1.0d0 + x))
    if (y <= 66000000.0d0) then
        tmp = (t_1 + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y))
    else
        tmp = (t_1 + (0.5d0 * (1.0d0 / sqrt(y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double tmp;
	if (y <= 66000000.0) {
		tmp = (t_1 + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (t_1 + (0.5 * (1.0 / Math.sqrt(y)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	tmp = 0
	if y <= 66000000.0:
		tmp = (t_1 + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (t_1 + (0.5 * (1.0 / math.sqrt(y)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 66000000.0)
		tmp = Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(t_1 + Float64(0.5 * Float64(1.0 / sqrt(y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	tmp = 0.0;
	if (y <= 66000000.0)
		tmp = (t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y));
	else
		tmp = (t_1 + (0.5 * (1.0 / sqrt(y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 66000000.0], N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6e7

   1. Initial program 97.4%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 60.7

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

   7. Applied rewrites 60.7%

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

   if 6.6e7 < y

   1. Initial program 77.8%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 81.1

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

   10. Applied rewrites 81.1%

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

Alternative 14: 33.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + x} + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.8e-44)
   (- (+ (sqrt (+ 1.0 y)) (sqrt (+ 1.0 z))) (sqrt y))
   (- (+ (sqrt (+ 1.0 x)) (* 0.5 (/ 1.0 (sqrt y)))) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.8e-44) {
		tmp = (sqrt((1.0 + y)) + sqrt((1.0 + z))) - sqrt(y);
	} else {
		tmp = (sqrt((1.0 + x)) + (0.5 * (1.0 / sqrt(y)))) - 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) :: tmp
    if (y <= 3.8d-44) then
        tmp = (sqrt((1.0d0 + y)) + sqrt((1.0d0 + z))) - sqrt(y)
    else
        tmp = (sqrt((1.0d0 + x)) + (0.5d0 * (1.0d0 / sqrt(y)))) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.8e-44) {
		tmp = (Math.sqrt((1.0 + y)) + Math.sqrt((1.0 + z))) - Math.sqrt(y);
	} else {
		tmp = (Math.sqrt((1.0 + x)) + (0.5 * (1.0 / Math.sqrt(y)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3.8e-44:
		tmp = (math.sqrt((1.0 + y)) + math.sqrt((1.0 + z))) - math.sqrt(y)
	else:
		tmp = (math.sqrt((1.0 + x)) + (0.5 * (1.0 / math.sqrt(y)))) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.8e-44)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + z))) - sqrt(y));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(0.5 * Float64(1.0 / sqrt(y)))) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.8e-44)
		tmp = (sqrt((1.0 + y)) + sqrt((1.0 + z))) - sqrt(y);
	else
		tmp = (sqrt((1.0 + x)) + (0.5 * (1.0 / sqrt(y)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 3.8e-44], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000001e-44

   1. Initial program 97.9%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in y around inf

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

      1. lift-sqrt.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in y around inf

      \[\leadsto \sqrt{y} - \sqrt{y} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 3.1

         \[\leadsto \sqrt{y} - \sqrt{y} \]

   10. Applied rewrites 3.1%

       \[\leadsto \sqrt{y} - \sqrt{y} \]

   11. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-+.f64 14.3

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

   13. Applied rewrites 14.3%

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

   if 3.8000000000000001e-44 < y

   1. Initial program 83.7%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 10.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 4.8%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 60.3

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

   10. Applied rewrites 60.3%

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

Alternative 15: 12.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{+32}:\\ \;\;\;\;\left(\sqrt{1 + x} + \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\sqrt{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.15e+32)
   (- (+ (sqrt (+ 1.0 x)) (sqrt z)) (+ (sqrt x) (sqrt z)))
   (* 0.5 (/ 1.0 (sqrt y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.15e+32) {
		tmp = (sqrt((1.0 + x)) + sqrt(z)) - (sqrt(x) + sqrt(z));
	} else {
		tmp = 0.5 * (1.0 / 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 (z <= 3.15d+32) then
        tmp = (sqrt((1.0d0 + x)) + sqrt(z)) - (sqrt(x) + sqrt(z))
    else
        tmp = 0.5d0 * (1.0d0 / sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.15e+32) {
		tmp = (Math.sqrt((1.0 + x)) + Math.sqrt(z)) - (Math.sqrt(x) + Math.sqrt(z));
	} else {
		tmp = 0.5 * (1.0 / Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3.15e+32:
		tmp = (math.sqrt((1.0 + x)) + math.sqrt(z)) - (math.sqrt(x) + math.sqrt(z))
	else:
		tmp = 0.5 * (1.0 / math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.15e+32)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + sqrt(z)) - Float64(sqrt(x) + sqrt(z)));
	else
		tmp = Float64(0.5 * Float64(1.0 / sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.15e+32)
		tmp = (sqrt((1.0 + x)) + sqrt(z)) - (sqrt(x) + sqrt(z));
	else
		tmp = 0.5 * (1.0 / sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 3.15e+32], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.1500000000000001e32

   1. Initial program 95.5%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 18.4

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

   10. Applied rewrites 18.4%

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

   if 3.1500000000000001e32 < z

   1. Initial program 89.5%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 3.2%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{y}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{y}} \]

      2. sqrt-div N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{y}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{1}{\sqrt{y}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{1}{\sqrt{y}} \]

      5. lift-/.f64 8.5

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

   10. Applied rewrites 8.5%

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

Alternative 16: 11.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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((1.0d0 + y)) + sqrt((1.0d0 + z))) - sqrt(y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in t around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in y around inf

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

   1. lift-sqrt.f64 1.5

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

7. Applied rewrites 1.5%

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

8. Taylor expanded in y around inf

   \[\leadsto \sqrt{y} - \sqrt{y} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 3.1

      \[\leadsto \sqrt{y} - \sqrt{y} \]

10. Applied rewrites 3.1%

    \[\leadsto \sqrt{y} - \sqrt{y} \]

11. Taylor expanded in x around inf

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

    1. lower-+.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-+.f64 11.3

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

13. Applied rewrites 11.3%

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

Alternative 17: 7.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

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 * (1.0d0 / sqrt(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in t around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

7. Applied rewrites 5.5%

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

8. Taylor expanded in y around 0

   \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{y}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{y}} \]

   2. sqrt-div N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{y}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\sqrt{y}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{1}{\sqrt{y}} \]

   5. lift-/.f64 7.9

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

10. Applied rewrites 7.9%

    \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\sqrt{y}}} \]
11. Add Preprocessing

Alternative 18: 7.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

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(z) - sqrt(y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in t around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in y around inf

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

   1. lift-sqrt.f64 1.5

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

7. Applied rewrites 1.5%

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

8. Taylor expanded in y around inf

   \[\leadsto \sqrt{y} - \sqrt{y} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 3.1

      \[\leadsto \sqrt{y} - \sqrt{y} \]

10. Applied rewrites 3.1%

    \[\leadsto \sqrt{y} - \sqrt{y} \]

11. Taylor expanded in z around inf

    \[\leadsto \sqrt{z} - \sqrt{\color{blue}{y}} \]
12. Step-by-step derivation

    1. lift-sqrt.f64 7.6

       \[\leadsto \sqrt{z} - \sqrt{y} \]

13. Applied rewrites 7.6%

    \[\leadsto \sqrt{z} - \sqrt{\color{blue}{y}} \]
14. Add Preprocessing

Alternative 19: 3.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

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(y) - sqrt(y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in t around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in y around inf

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

   1. lift-sqrt.f64 1.5

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

7. Applied rewrites 1.5%

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

8. Taylor expanded in y around inf

   \[\leadsto \sqrt{y} - \sqrt{y} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 3.1

      \[\leadsto \sqrt{y} - \sqrt{y} \]

10. Applied rewrites 3.1%

    \[\leadsto \sqrt{y} - \sqrt{y} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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))))