Main:z from

Percentage Accurate: 91.8% → 98.0%

Time: 16.7s

Alternatives: 24

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: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 4600000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.5, \sqrt{x}, 0.5 \cdot \left(x \cdot {y}^{-0.5}\right)\right)}{x} + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= x 4600000000000.0)
     (+
      (+
       (+ (- (sqrt (+ x 1.0)) (sqrt x)) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
       (- (sqrt (+ z 1.0)) (sqrt z)))
      t_1)
     (+
      (+
       (/ (fma 0.5 (sqrt x) (* 0.5 (* x (pow y -0.5)))) x)
       (* 0.5 (/ 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 (x <= 4600000000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	} else {
		tmp = ((fma(0.5, sqrt(x), (0.5 * (x * pow(y, -0.5)))) / x) + (0.5 * (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 (x <= 4600000000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_1);
	else
		tmp = Float64(Float64(Float64(fma(0.5, sqrt(x), Float64(0.5 * Float64(x * (y ^ -0.5)))) / x) + Float64(0.5 * Float64(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[x, 4600000000000.0], N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $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[Sqrt[x], $MachinePrecision] + N[(0.5 * N[(x * N[Power[y, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e12

   1. Initial program 96.7%

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

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 97.8%

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

   if 4.6e12 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 46.2

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

   8. Applied rewrites 46.2%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 23.6

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

   11. Applied rewrites 23.6%

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

   12. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sqrt-div N/A

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

       7. metadata-eval N/A

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

       8. unpow-1 N/A

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

       9. sqrt-pow2 N/A

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

       10. metadata-eval N/A

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

       11. lift-pow.f64 23.3

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

   14. Applied rewrites 23.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4600000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.5, \sqrt{x}, 0.5 \cdot \left(x \cdot {y}^{-0.5}\right)\right)}{x} + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(t\_2 + \left(t\_1 - \sqrt{z}\right)\right) + t\_3\\ t_5 := \frac{1}{t\_1 + \sqrt{z}}\\ \mathbf{if}\;t\_4 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 3.0002:\\ \;\;\;\;\left(t\_2 + t\_5\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_5\right) + t\_3\\ \end{array} \end{array} \]

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)))
        (t_2 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (+ (+ t_2 (- t_1 (sqrt z))) t_3))
        (t_5 (/ 1.0 (+ t_1 (sqrt z)))))
   (if (<= t_4 4e-7)
     (+
      (+
       (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
       (* 0.5 (/ 1.0 (sqrt z))))
      t_3)
     (if (<= t_4 3.0002)
       (+ (+ t_2 t_5) (* 0.5 (/ 1.0 (sqrt t))))
       (+ (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_5) 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));
	double t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (t_2 + (t_1 - sqrt(z))) + t_3;
	double t_5 = 1.0 / (t_1 + sqrt(z));
	double tmp;
	if (t_4 <= 4e-7) {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (0.5 * (1.0 / sqrt(z)))) + t_3;
	} else if (t_4 <= 3.0002) {
		tmp = (t_2 + t_5) + (0.5 * (1.0 / sqrt(t)));
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_5) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(t_2 + Float64(t_1 - sqrt(z))) + t_3)
	t_5 = Float64(1.0 / Float64(t_1 + sqrt(z)))
	tmp = 0.0
	if (t_4 <= 4e-7)
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(0.5 * Float64(1.0 / sqrt(z)))) + t_3);
	elseif (t_4 <= 3.0002)
		tmp = Float64(Float64(t_2 + t_5) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_5) + 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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 4e-7], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 3.0002], N[(N[(t$95$2 + t$95$5), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $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))) < 3.9999999999999998e-7

   1. Initial program 5.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 71.1

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

   11. Applied rewrites 71.1%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 71.1

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

   13. Applied rewrites 71.1%

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

   if 3.9999999999999998e-7 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.00019999999999998

   1. Initial program 95.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 54.0

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

   10. Applied rewrites 54.0%

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

   if 3.00019999999999998 < (+.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 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 95.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 92.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 3.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{z + 1}\\ t_3 := \frac{1}{\sqrt{y}}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(t\_2 - \sqrt{z}\right)\right) + t\_1\\ t_5 := 0.5 \cdot \frac{1}{\sqrt{z}}\\ \mathbf{if}\;t\_4 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot t\_3\right) + t\_5\right) + t\_1\\ \mathbf{elif}\;t\_4 \leq 1.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(t\_3, 0.5, \sqrt{1 + x}\right) - \sqrt{x}\right) + t\_5\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \frac{1}{t\_2 + \sqrt{z}}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (/ 1.0 (sqrt y)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- t_2 (sqrt z)))
          t_1))
        (t_5 (* 0.5 (/ 1.0 (sqrt z)))))
   (if (<= t_4 4e-7)
     (+ (+ (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 t_3)) t_5) t_1)
     (if (<= t_4 1.5)
       (+ (+ (- (fma t_3 0.5 (sqrt (+ 1.0 x))) (sqrt x)) t_5) t_1)
       (+
        (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) (/ 1.0 (+ t_2 (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 t_2 = sqrt((z + 1.0));
	double t_3 = 1.0 / sqrt(y);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (t_2 - sqrt(z))) + t_1;
	double t_5 = 0.5 * (1.0 / sqrt(z));
	double tmp;
	if (t_4 <= 4e-7) {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * t_3)) + t_5) + t_1;
	} else if (t_4 <= 1.5) {
		tmp = ((fma(t_3, 0.5, sqrt((1.0 + x))) - sqrt(x)) + t_5) + t_1;
	} else {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + (1.0 / (t_2 + 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))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(1.0 / sqrt(y))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(t_2 - sqrt(z))) + t_1)
	t_5 = Float64(0.5 * Float64(1.0 / sqrt(z)))
	tmp = 0.0
	if (t_4 <= 4e-7)
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * t_3)) + t_5) + t_1);
	elseif (t_4 <= 1.5)
		tmp = Float64(Float64(Float64(fma(t_3, 0.5, sqrt(Float64(1.0 + x))) - sqrt(x)) + t_5) + t_1);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + Float64(1.0 / Float64(t_2 + 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]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[Sqrt[y], $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] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 4e-7], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.5], N[(N[(N[(N[(t$95$3 * 0.5 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $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))) < 3.9999999999999998e-7

   1. Initial program 5.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 71.1

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

   11. Applied rewrites 71.1%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 71.1

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

   13. Applied rewrites 71.1%

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

   if 3.9999999999999998e-7 < (+.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))) < 1.5

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 65.1

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

   5. Applied rewrites 65.1%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left({\left(\sqrt{y}\right)}^{-1}, 0.5, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 48.0

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

   10. Applied rewrites 48.0%

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

   if 1.5 < (+.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 97.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 59.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 35.8%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1.5:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, \sqrt{1 + x}\right) - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(t\_2 - \sqrt{z}\right)\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 1:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{x}\right) + \frac{1}{t\_2 + \sqrt{z}}\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 2.0005:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{1 + y} + 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_1 + 1\right) + t\_2\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- t_2 (sqrt z)))
          t_3)))
   (if (<= t_4 1.0)
     (+ (+ (- t_1 (sqrt x)) (/ 1.0 (+ t_2 (sqrt z)))) t_3)
     (if (<= t_4 2.0005)
       (-
        (+ t_1 (+ (sqrt (+ 1.0 y)) (* 0.5 (/ 1.0 (sqrt z)))))
        (+ (sqrt x) (sqrt y)))
       (- (- (+ (+ t_1 1.0) t_2) (sqrt x)) (+ (sqrt z) (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((z + 1.0));
	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 - sqrt(z))) + t_3;
	double tmp;
	if (t_4 <= 1.0) {
		tmp = ((t_1 - sqrt(x)) + (1.0 / (t_2 + sqrt(z)))) + t_3;
	} else if (t_4 <= 2.0005) {
		tmp = (t_1 + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((t_1 + 1.0) + t_2) - sqrt(x)) - (sqrt(z) + 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) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (t_2 - sqrt(z))) + t_3
    if (t_4 <= 1.0d0) then
        tmp = ((t_1 - sqrt(x)) + (1.0d0 / (t_2 + sqrt(z)))) + t_3
    else if (t_4 <= 2.0005d0) then
        tmp = (t_1 + (sqrt((1.0d0 + y)) + (0.5d0 * (1.0d0 / sqrt(z))))) - (sqrt(x) + sqrt(y))
    else
        tmp = (((t_1 + 1.0d0) + t_2) - sqrt(x)) - (sqrt(z) + 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((z + 1.0));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (t_2 - Math.sqrt(z))) + t_3;
	double tmp;
	if (t_4 <= 1.0) {
		tmp = ((t_1 - Math.sqrt(x)) + (1.0 / (t_2 + Math.sqrt(z)))) + t_3;
	} else if (t_4 <= 2.0005) {
		tmp = (t_1 + (Math.sqrt((1.0 + y)) + (0.5 * (1.0 / Math.sqrt(z))))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((t_1 + 1.0) + t_2) - Math.sqrt(x)) - (Math.sqrt(z) + 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((z + 1.0))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (t_2 - math.sqrt(z))) + t_3
	tmp = 0
	if t_4 <= 1.0:
		tmp = ((t_1 - math.sqrt(x)) + (1.0 / (t_2 + math.sqrt(z)))) + t_3
	elif t_4 <= 2.0005:
		tmp = (t_1 + (math.sqrt((1.0 + y)) + (0.5 * (1.0 / math.sqrt(z))))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((t_1 + 1.0) + t_2) - math.sqrt(x)) - (math.sqrt(z) + 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 = sqrt(Float64(z + 1.0))
	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))) + Float64(t_2 - sqrt(z))) + t_3)
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(1.0 / Float64(t_2 + sqrt(z)))) + t_3);
	elseif (t_4 <= 2.0005)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(1.0 / sqrt(z))))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 + 1.0) + t_2) - sqrt(x)) - Float64(sqrt(z) + 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((z + 1.0));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (t_2 - sqrt(z))) + t_3;
	tmp = 0.0;
	if (t_4 <= 1.0)
		tmp = ((t_1 - sqrt(x)) + (1.0 / (t_2 + sqrt(z)))) + t_3;
	elseif (t_4 <= 2.0005)
		tmp = (t_1 + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	else
		tmp = (((t_1 + 1.0) + t_2) - sqrt(x)) - (sqrt(z) + 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[Sqrt[N[(z + 1.0), $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] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 2.0005], N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 79.8%

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

   8. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 61.5

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

   10. Applied rewrites 61.5%

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

   if 1 < (+.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))) < 2.00050000000000017

   1. Initial program 96.0%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 19.8%

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

   if 2.00050000000000017 < (+.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 98.0%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 24.8%

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

Alternative 5: 65.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ t_3 := t\_1 + t\_2\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_4 \leq 1.9999999999999987:\\ \;\;\;\;t\_3 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;\left(t\_1 + \left(t\_2 + 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_3\right) - \sqrt{x}\right) - \sqrt{y}\\ \end{array} \end{array} \]

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 (+ 1.0 y)))
        (t_3 (+ t_1 t_2))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_4 1.9999999999999987)
     (- t_3 (+ (sqrt x) (sqrt y)))
     (if (<= t_4 2.5)
       (- (+ t_1 (+ t_2 (* 0.5 (/ 1.0 (sqrt z))))) (sqrt x))
       (- (- (+ 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((1.0 + y));
	double t_3 = t_1 + t_2;
	double t_4 = (((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 tmp;
	if (t_4 <= 1.9999999999999987) {
		tmp = t_3 - (sqrt(x) + sqrt(y));
	} else if (t_4 <= 2.5) {
		tmp = (t_1 + (t_2 + (0.5 * (1.0 / sqrt(z))))) - sqrt(x);
	} 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) :: t_4
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + y))
    t_3 = t_1 + t_2
    t_4 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
    if (t_4 <= 1.9999999999999987d0) then
        tmp = t_3 - (sqrt(x) + sqrt(y))
    else if (t_4 <= 2.5d0) then
        tmp = (t_1 + (t_2 + (0.5d0 * (1.0d0 / sqrt(z))))) - sqrt(x)
    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((1.0 + y));
	double t_3 = t_1 + t_2;
	double t_4 = (((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 tmp;
	if (t_4 <= 1.9999999999999987) {
		tmp = t_3 - (Math.sqrt(x) + Math.sqrt(y));
	} else if (t_4 <= 2.5) {
		tmp = (t_1 + (t_2 + (0.5 * (1.0 / Math.sqrt(z))))) - Math.sqrt(x);
	} 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((1.0 + y))
	t_3 = t_1 + t_2
	t_4 = (((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))
	tmp = 0
	if t_4 <= 1.9999999999999987:
		tmp = t_3 - (math.sqrt(x) + math.sqrt(y))
	elif t_4 <= 2.5:
		tmp = (t_1 + (t_2 + (0.5 * (1.0 / math.sqrt(z))))) - math.sqrt(x)
	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 = sqrt(Float64(1.0 + y))
	t_3 = Float64(t_1 + t_2)
	t_4 = 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)))
	tmp = 0.0
	if (t_4 <= 1.9999999999999987)
		tmp = Float64(t_3 - Float64(sqrt(x) + sqrt(y)));
	elseif (t_4 <= 2.5)
		tmp = Float64(Float64(t_1 + Float64(t_2 + Float64(0.5 * Float64(1.0 / sqrt(z))))) - sqrt(x));
	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((1.0 + y));
	t_3 = t_1 + t_2;
	t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	tmp = 0.0;
	if (t_4 <= 1.9999999999999987)
		tmp = t_3 - (sqrt(x) + sqrt(y));
	elseif (t_4 <= 2.5)
		tmp = (t_1 + (t_2 + (0.5 * (1.0 / sqrt(z))))) - sqrt(x);
	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[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + t$95$2), $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] + 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]}, If[LessEqual[t$95$4, 1.9999999999999987], N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(N[(t$95$1 + N[(t$95$2 + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $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))) < 1.9999999999999987

   1. Initial program 77.1%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. 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 10.5

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

   8. Applied rewrites 10.5%

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

   if 1.9999999999999987 < (+.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 96.7%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in x around inf

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

       1. lift-sqrt.f64 20.8

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

   11. Applied rewrites 20.8%

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

   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 98.1%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 2.2

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in y around inf

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

       1. lift-sqrt.f64 4.6

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

   11. Applied rewrites 4.6%

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

   12. Taylor expanded in z around 0

       \[\leadsto \left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \sqrt{\color{blue}{y}} \]
   13. 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. lift-sqrt.f64 N/A

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

       7. lift-+.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 36.6

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

   14. Applied rewrites 36.6%

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

Alternative 6: 86.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_2\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 \leq 1:\\ \;\;\;\;\left(\left(t\_4 - \sqrt{x}\right) + t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_3 \leq 2.0005:\\ \;\;\;\;\left(t\_4 + \left(\sqrt{1 + y} + 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_4 + 1\right) + t\_1\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (- t_1 (sqrt z)))
        (t_3
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_2))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= t_3 1.0)
     (+ (+ (- t_4 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_3 2.0005)
       (-
        (+ t_4 (+ (sqrt (+ 1.0 y)) (* 0.5 (/ 1.0 (sqrt z)))))
        (+ (sqrt x) (sqrt y)))
       (- (- (+ (+ t_4 1.0) t_1) (sqrt x)) (+ (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = t_1 - sqrt(z);
	double t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2;
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 1.0) {
		tmp = ((t_4 - sqrt(x)) + t_2) + (sqrt(t) - sqrt(t));
	} else if (t_3 <= 2.0005) {
		tmp = (t_4 + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((t_4 + 1.0) + t_1) - sqrt(x)) - (sqrt(z) + 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) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = t_1 - sqrt(z)
    t_3 = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_2
    t_4 = sqrt((1.0d0 + x))
    if (t_3 <= 1.0d0) then
        tmp = ((t_4 - sqrt(x)) + t_2) + (sqrt(t) - sqrt(t))
    else if (t_3 <= 2.0005d0) then
        tmp = (t_4 + (sqrt((1.0d0 + y)) + (0.5d0 * (1.0d0 / sqrt(z))))) - (sqrt(x) + sqrt(y))
    else
        tmp = (((t_4 + 1.0d0) + t_1) - sqrt(x)) - (sqrt(z) + sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = t_1 - Math.sqrt(z);
	double t_3 = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_2;
	double t_4 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 1.0) {
		tmp = ((t_4 - Math.sqrt(x)) + t_2) + (Math.sqrt(t) - Math.sqrt(t));
	} else if (t_3 <= 2.0005) {
		tmp = (t_4 + (Math.sqrt((1.0 + y)) + (0.5 * (1.0 / Math.sqrt(z))))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((t_4 + 1.0) + t_1) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = t_1 - math.sqrt(z)
	t_3 = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_2
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_3 <= 1.0:
		tmp = ((t_4 - math.sqrt(x)) + t_2) + (math.sqrt(t) - math.sqrt(t))
	elif t_3 <= 2.0005:
		tmp = (t_4 + (math.sqrt((1.0 + y)) + (0.5 * (1.0 / math.sqrt(z))))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((t_4 + 1.0) + t_1) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_2)
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_3 <= 1.0)
		tmp = Float64(Float64(Float64(t_4 - sqrt(x)) + t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_3 <= 2.0005)
		tmp = Float64(Float64(t_4 + Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(1.0 / sqrt(z))))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(t_4 + 1.0) + t_1) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = t_1 - sqrt(z);
	t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2;
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_3 <= 1.0)
		tmp = ((t_4 - sqrt(x)) + t_2) + (sqrt(t) - sqrt(t));
	elseif (t_3 <= 2.0005)
		tmp = (t_4 + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	else
		tmp = (((t_4 + 1.0) + t_1) - sqrt(x)) - (sqrt(z) + sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1.0], N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0005], N[(N[(t$95$4 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 83.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 29.2

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

   8. Applied rewrites 29.2%

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

   if 1 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00050000000000017

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 24.5%

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

   if 2.00050000000000017 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.4%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.4%

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

Alternative 7: 86.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_2\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 \leq 1:\\ \;\;\;\;\left(\left(t\_4 - \sqrt{x}\right) + t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_3 \leq 2.0005:\\ \;\;\;\;\left(\left(1 + 0.5 \cdot x\right) + \left(\sqrt{1 + y} + 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_4 + 1\right) + t\_1\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (- t_1 (sqrt z)))
        (t_3
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_2))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= t_3 1.0)
     (+ (+ (- t_4 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_3 2.0005)
       (-
        (+ (+ 1.0 (* 0.5 x)) (+ (sqrt (+ 1.0 y)) (* 0.5 (/ 1.0 (sqrt z)))))
        (+ (sqrt x) (sqrt y)))
       (- (- (+ (+ t_4 1.0) t_1) (sqrt x)) (+ (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = t_1 - sqrt(z);
	double t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2;
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 1.0) {
		tmp = ((t_4 - sqrt(x)) + t_2) + (sqrt(t) - sqrt(t));
	} else if (t_3 <= 2.0005) {
		tmp = ((1.0 + (0.5 * x)) + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((t_4 + 1.0) + t_1) - sqrt(x)) - (sqrt(z) + 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) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = t_1 - sqrt(z)
    t_3 = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_2
    t_4 = sqrt((1.0d0 + x))
    if (t_3 <= 1.0d0) then
        tmp = ((t_4 - sqrt(x)) + t_2) + (sqrt(t) - sqrt(t))
    else if (t_3 <= 2.0005d0) then
        tmp = ((1.0d0 + (0.5d0 * x)) + (sqrt((1.0d0 + y)) + (0.5d0 * (1.0d0 / sqrt(z))))) - (sqrt(x) + sqrt(y))
    else
        tmp = (((t_4 + 1.0d0) + t_1) - sqrt(x)) - (sqrt(z) + sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = t_1 - Math.sqrt(z);
	double t_3 = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_2;
	double t_4 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_3 <= 1.0) {
		tmp = ((t_4 - Math.sqrt(x)) + t_2) + (Math.sqrt(t) - Math.sqrt(t));
	} else if (t_3 <= 2.0005) {
		tmp = ((1.0 + (0.5 * x)) + (Math.sqrt((1.0 + y)) + (0.5 * (1.0 / Math.sqrt(z))))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((t_4 + 1.0) + t_1) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = t_1 - math.sqrt(z)
	t_3 = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_2
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_3 <= 1.0:
		tmp = ((t_4 - math.sqrt(x)) + t_2) + (math.sqrt(t) - math.sqrt(t))
	elif t_3 <= 2.0005:
		tmp = ((1.0 + (0.5 * x)) + (math.sqrt((1.0 + y)) + (0.5 * (1.0 / math.sqrt(z))))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((t_4 + 1.0) + t_1) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_2)
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_3 <= 1.0)
		tmp = Float64(Float64(Float64(t_4 - sqrt(x)) + t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_3 <= 2.0005)
		tmp = Float64(Float64(Float64(1.0 + Float64(0.5 * x)) + Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(1.0 / sqrt(z))))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(t_4 + 1.0) + t_1) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = t_1 - sqrt(z);
	t_3 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2;
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_3 <= 1.0)
		tmp = ((t_4 - sqrt(x)) + t_2) + (sqrt(t) - sqrt(t));
	elseif (t_3 <= 2.0005)
		tmp = ((1.0 + (0.5 * x)) + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	else
		tmp = (((t_4 + 1.0) + t_1) - sqrt(x)) - (sqrt(z) + sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1.0], N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0005], N[(N[(N[(1.0 + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 83.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 29.2

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

   8. Applied rewrites 29.2%

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

   if 1 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00050000000000017

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 24.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 22.7

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

   11. Applied rewrites 22.7%

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

   if 2.00050000000000017 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.4%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.4%

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

Alternative 8: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + x} - \left(\sqrt{x} - t\_3\right)\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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) 4e-7)
     (+
      (+
       (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
       (* 0.5 (/ 1.0 (sqrt z))))
      t_2)
     (+ (+ (- (sqrt (+ 1.0 x)) (- (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) <= 4e-7) {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (0.5 * (1.0 / sqrt(z)))) + t_2;
	} else {
		tmp = ((sqrt((1.0 + x)) - (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) <= 4e-7)
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(0.5 * Float64(1.0 / sqrt(z)))) + t_2);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) - Float64(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], 4e-7], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] - t$95$3), $MachinePrecision]), $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))) < 3.9999999999999998e-7

   1. Initial program 5.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 71.1

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

   11. Applied rewrites 71.1%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 71.1

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

   13. Applied rewrites 71.1%

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

   if 3.9999999999999998e-7 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 96.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-+l- N/A

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

      11. +-commutative N/A

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

      12. lower--.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-sqrt.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

   4. Applied rewrites 77.3%

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

Alternative 9: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          t_1)))
   (if (<= t_2 4e-7)
     (+
      (+
       (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
       (* 0.5 (/ 1.0 (sqrt z))))
      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((t + 1.0)) - sqrt(t);
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	double tmp;
	if (t_2 <= 4e-7) {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (0.5 * (1.0 / sqrt(z)))) + t_1;
	} else {
		tmp = 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(t + 1.0)) - sqrt(t))
	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))) + t_1)
	tmp = 0.0
	if (t_2 <= 4e-7)
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(0.5 * Float64(1.0 / sqrt(z)))) + t_1);
	else
		tmp = 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[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $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] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-7], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]

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))) < 3.9999999999999998e-7

   1. Initial program 5.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 71.1

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

   11. Applied rewrites 71.1%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 71.1

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

   13. Applied rewrites 71.1%

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

   if 3.9999999999999998e-7 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 96.3%

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

Alternative 10: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2 \leq 0.4:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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) 0.4)
     (+
      (+
       (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
       (* 0.5 (/ 1.0 (sqrt z))))
      t_2)
     (+ (+ (+ (- (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) <= 0.4) {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (0.5 * (1.0 / sqrt(z)))) + t_2;
	} 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) <= 0.4)
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(0.5 * Float64(1.0 / sqrt(z)))) + t_2);
	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], 0.4], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $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))) < 0.40000000000000002

   1. Initial program 10.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 39.9

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

   8. Applied rewrites 39.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 68.5

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

   11. Applied rewrites 68.5%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 68.5

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

   13. Applied rewrites 68.5%

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

   if 0.40000000000000002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. 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 54.1

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

   5. Applied rewrites 54.1%

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

4. Final simplification 55.1%

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

Alternative 11: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2 \leq 0.4:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{\sqrt{x}} + t\_3\right) + t\_1\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) + t\_3\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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) 0.4)
     (+ (+ (+ (* 0.5 (/ 1.0 (sqrt x))) t_3) t_1) (- (sqrt t) (sqrt t)))
     (+ (+ (+ (- (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) <= 0.4) {
		tmp = (((0.5 * (1.0 / sqrt(x))) + t_3) + t_1) + (sqrt(t) - sqrt(t));
	} 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) <= 0.4)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + t_3) + t_1) + Float64(sqrt(t) - sqrt(t)));
	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], 0.4], N[(N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $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))) < 0.40000000000000002

   1. Initial program 10.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 24.5

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

   8. Applied rewrites 24.5%

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

   if 0.40000000000000002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. 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 54.1

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

   5. Applied rewrites 54.1%

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

4. Final simplification 52.1%

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

Alternative 12: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(t\_2 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 2.0005:\\ \;\;\;\;\left(t\_1 + \left(\sqrt{1 + y} + 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_1 + 1\right) + t\_2\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x))) (t_2 (sqrt (+ z 1.0))))
   (if (<=
        (+
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- t_2 (sqrt z)))
         (- (sqrt (+ t 1.0)) (sqrt t)))
        2.0005)
     (- (+ t_1 (+ (sqrt (+ 1.0 y)) (* 0.5 (/ 1.0 (sqrt z))))) (sqrt y))
     (- (- (+ (+ t_1 1.0) t_2) (sqrt x)) (+ (sqrt z) (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((z + 1.0));
	double tmp;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (t_2 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 2.0005) {
		tmp = (t_1 + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - sqrt(y);
	} else {
		tmp = (((t_1 + 1.0) + t_2) - sqrt(x)) - (sqrt(z) + 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) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((z + 1.0d0))
    if (((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (t_2 - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))) <= 2.0005d0) then
        tmp = (t_1 + (sqrt((1.0d0 + y)) + (0.5d0 * (1.0d0 / sqrt(z))))) - sqrt(y)
    else
        tmp = (((t_1 + 1.0d0) + t_2) - sqrt(x)) - (sqrt(z) + 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((z + 1.0));
	double tmp;
	if (((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (t_2 - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t))) <= 2.0005) {
		tmp = (t_1 + (Math.sqrt((1.0 + y)) + (0.5 * (1.0 / Math.sqrt(z))))) - Math.sqrt(y);
	} else {
		tmp = (((t_1 + 1.0) + t_2) - Math.sqrt(x)) - (Math.sqrt(z) + 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((z + 1.0))
	tmp = 0
	if ((((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (t_2 - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))) <= 2.0005:
		tmp = (t_1 + (math.sqrt((1.0 + y)) + (0.5 * (1.0 / math.sqrt(z))))) - math.sqrt(y)
	else:
		tmp = (((t_1 + 1.0) + t_2) - math.sqrt(x)) - (math.sqrt(z) + 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 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(t_2 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t))) <= 2.0005)
		tmp = Float64(Float64(t_1 + Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(1.0 / sqrt(z))))) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 + 1.0) + t_2) - sqrt(x)) - Float64(sqrt(z) + 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((z + 1.0));
	tmp = 0.0;
	if (((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (t_2 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t))) <= 2.0005)
		tmp = (t_1 + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - sqrt(y);
	else
		tmp = (((t_1 + 1.0) + t_2) - sqrt(x)) - (sqrt(z) + 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[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0005], N[(N[(t$95$1 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00050000000000017

   1. Initial program 86.9%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 15.3%

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

   9. Taylor expanded in y around inf

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

       1. lift-sqrt.f64 14.6

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

   11. Applied rewrites 14.6%

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

   if 2.00050000000000017 < (+.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 98.0%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 24.8%

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

Alternative 13: 64.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} + \sqrt{1 + y}\\ \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right) \leq 2.5:\\ \;\;\;\;t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_1\right) - \sqrt{x}\right) - \sqrt{y}\\ \end{array} \end{array} \]

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)) (sqrt (+ 1.0 y)))))
   (if (<=
        (+
         (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
         (- (sqrt (+ z 1.0)) (sqrt z)))
        2.5)
     (- t_1 (+ (sqrt x) (sqrt y)))
     (- (- (+ 1.0 t_1) (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)) + sqrt((1.0 + y));
	double tmp;
	if ((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) <= 2.5) {
		tmp = t_1 - (sqrt(x) + sqrt(y));
	} else {
		tmp = ((1.0 + t_1) - 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) :: tmp
    t_1 = sqrt((1.0d0 + x)) + sqrt((1.0d0 + y))
    if ((((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) <= 2.5d0) then
        tmp = t_1 - (sqrt(x) + sqrt(y))
    else
        tmp = ((1.0d0 + t_1) - 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)) + Math.sqrt((1.0 + y));
	double tmp;
	if ((((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) <= 2.5) {
		tmp = t_1 - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = ((1.0 + t_1) - 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)) + math.sqrt((1.0 + y))
	tmp = 0
	if (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) <= 2.5:
		tmp = t_1 - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = ((1.0 + t_1) - 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 = Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 + y)))
	tmp = 0.0
	if (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))) <= 2.5)
		tmp = Float64(t_1 - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(1.0 + t_1) - 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)) + sqrt((1.0 + y));
	tmp = 0.0;
	if ((((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) <= 2.5)
		tmp = t_1 - (sqrt(x) + sqrt(y));
	else
		tmp = ((1.0 + t_1) - 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[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.5], N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.5

   1. Initial program 88.7%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. 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 14.2

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

   8. Applied rewrites 14.2%

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

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 2.4

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in y around inf

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

       1. lift-sqrt.f64 5.2

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

   11. Applied rewrites 5.2%

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

   12. Taylor expanded in z around 0

       \[\leadsto \left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \sqrt{\color{blue}{y}} \]
   13. 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. lift-sqrt.f64 N/A

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

       7. lift-+.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 45.1

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

   14. Applied rewrites 45.1%

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

Alternative 14: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 4600000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= x 4600000000000.0)
     (+
      (+
       (+ (- (sqrt (+ x 1.0)) (sqrt x)) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
       (- (sqrt (+ z 1.0)) (sqrt z)))
      t_1)
     (+
      (+
       (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
       (* 0.5 (/ 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 (x <= 4600000000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	} else {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (0.5 * (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 (x <= 4600000000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_1);
	else
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(0.5 * Float64(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[x, 4600000000000.0], N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e12

   1. Initial program 96.7%

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

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 97.8%

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

   if 4.6e12 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 46.2

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

   8. Applied rewrites 46.2%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 23.6

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

   11. Applied rewrites 23.6%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 23.6

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

   13. Applied rewrites 23.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4600000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 4600000000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + 0.5 \cdot \frac{1}{\sqrt{z}}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= x 4600000000000.0)
     (+
      (+
       (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
       (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
      t_1)
     (+
      (+
       (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
       (* 0.5 (/ 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 (x <= 4600000000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + t_1;
	} else {
		tmp = (fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (0.5 * (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 (x <= 4600000000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + t_1);
	else
		tmp = Float64(Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(0.5 * Float64(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[x, 4600000000000.0], N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e12

   1. Initial program 96.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 97.1%

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

   if 4.6e12 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. unpow-1 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 46.2

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

   8. Applied rewrites 46.2%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-sqrt.f64 23.6

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

   11. Applied rewrites 23.6%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. sqrt-pow2 N/A

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

       4. unpow-1 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-sqrt.f64 23.6

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

   13. Applied rewrites 23.6%

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

Alternative 16: 90.5% accurate, 0.9× speedup

Math

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

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.5

   1. Initial program 83.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 82.8

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

   5. Applied rewrites 82.8%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left({\left(\sqrt{y}\right)}^{-1}, 0.5, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 46.7%

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

   if 0.5 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 97.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 96.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 50.4%

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

4. Final simplification 48.5%

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

Alternative 17: 69.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0))))
   (if (<= (- t_1 (sqrt z)) 2e-5)
     (-
      (+ (+ 1.0 (* 0.5 x)) (+ (sqrt (+ 1.0 y)) (* 0.5 (/ 1.0 (sqrt z)))))
      (+ (sqrt x) (sqrt y)))
     (- (- (+ (+ (sqrt (+ 1.0 x)) 1.0) t_1) (sqrt x)) (+ (sqrt z) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double tmp;
	if ((t_1 - sqrt(z)) <= 2e-5) {
		tmp = ((1.0 + (0.5 * x)) + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	} else {
		tmp = (((sqrt((1.0 + x)) + 1.0) + t_1) - sqrt(x)) - (sqrt(z) + 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) :: tmp
    t_1 = sqrt((z + 1.0d0))
    if ((t_1 - sqrt(z)) <= 2d-5) then
        tmp = ((1.0d0 + (0.5d0 * x)) + (sqrt((1.0d0 + y)) + (0.5d0 * (1.0d0 / sqrt(z))))) - (sqrt(x) + sqrt(y))
    else
        tmp = (((sqrt((1.0d0 + x)) + 1.0d0) + t_1) - sqrt(x)) - (sqrt(z) + sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(z)) <= 2e-5) {
		tmp = ((1.0 + (0.5 * x)) + (Math.sqrt((1.0 + y)) + (0.5 * (1.0 / Math.sqrt(z))))) - (Math.sqrt(x) + Math.sqrt(y));
	} else {
		tmp = (((Math.sqrt((1.0 + x)) + 1.0) + t_1) - Math.sqrt(x)) - (Math.sqrt(z) + Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(z)) <= 2e-5:
		tmp = ((1.0 + (0.5 * x)) + (math.sqrt((1.0 + y)) + (0.5 * (1.0 / math.sqrt(z))))) - (math.sqrt(x) + math.sqrt(y))
	else:
		tmp = (((math.sqrt((1.0 + x)) + 1.0) + t_1) - math.sqrt(x)) - (math.sqrt(z) + math.sqrt(y))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 2e-5)
		tmp = Float64(Float64(Float64(1.0 + Float64(0.5 * x)) + Float64(sqrt(Float64(1.0 + y)) + Float64(0.5 * Float64(1.0 / sqrt(z))))) - Float64(sqrt(x) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + 1.0) + t_1) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(z)) <= 2e-5)
		tmp = ((1.0 + (0.5 * x)) + (sqrt((1.0 + y)) + (0.5 * (1.0 / sqrt(z))))) - (sqrt(x) + sqrt(y));
	else
		tmp = (((sqrt((1.0 + x)) + 1.0) + t_1) - sqrt(x)) - (sqrt(z) + sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 2.00000000000000016e-5

   1. Initial program 82.1%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 20.7

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

   11. Applied rewrites 20.7%

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

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 17.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 15.0%

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

Alternative 18: 91.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (((fma(0.5, x, 1.0) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Julia

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

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[(N[(N[(0.5 * x + 1.0), $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]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(\left(\color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. 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 50.7

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

5. Applied rewrites 50.7%

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

6. Final simplification 50.7%

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

Alternative 19: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.05:\\ \;\;\;\;\left(t\_2 + \left(t\_1 + 0.5 \cdot \frac{1}{\sqrt{z}}\right)\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(t\_2 + t\_1\right)\right) - \sqrt{x}\right) - \sqrt{y}\\ \end{array} \end{array} \]

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 y))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.05)
     (- (+ t_2 (+ t_1 (* 0.5 (/ 1.0 (sqrt z))))) (sqrt y))
     (- (- (+ 1.0 (+ t_2 t_1)) (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 + y));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.05) {
		tmp = (t_2 + (t_1 + (0.5 * (1.0 / sqrt(z))))) - sqrt(y);
	} else {
		tmp = ((1.0 + (t_2 + t_1)) - 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) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + x))
    if ((sqrt((z + 1.0d0)) - sqrt(z)) <= 0.05d0) then
        tmp = (t_2 + (t_1 + (0.5d0 * (1.0d0 / sqrt(z))))) - sqrt(y)
    else
        tmp = ((1.0d0 + (t_2 + t_1)) - 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 + y));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) <= 0.05) {
		tmp = (t_2 + (t_1 + (0.5 * (1.0 / Math.sqrt(z))))) - Math.sqrt(y);
	} else {
		tmp = ((1.0 + (t_2 + t_1)) - 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 + y))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if (math.sqrt((z + 1.0)) - math.sqrt(z)) <= 0.05:
		tmp = (t_2 + (t_1 + (0.5 * (1.0 / math.sqrt(z))))) - math.sqrt(y)
	else:
		tmp = ((1.0 + (t_2 + t_1)) - 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 + y))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) <= 0.05)
		tmp = Float64(Float64(t_2 + Float64(t_1 + Float64(0.5 * Float64(1.0 / sqrt(z))))) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(t_2 + t_1)) - 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 + y));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.05)
		tmp = (t_2 + (t_1 + (0.5 * (1.0 / sqrt(z))))) - sqrt(y);
	else
		tmp = ((1.0 + (t_2 + t_1)) - 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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(t$95$2 + N[(t$95$1 + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.050000000000000003

   1. Initial program 82.3%

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

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 22.4%

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

   9. Taylor expanded in y around inf

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

       1. lift-sqrt.f64 21.1

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

   11. Applied rewrites 21.1%

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

   if 0.050000000000000003 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 1.8

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

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in y around inf

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

       1. lift-sqrt.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in z around 0

       \[\leadsto \left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \sqrt{x}\right) - \sqrt{\color{blue}{y}} \]
   13. 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. lift-sqrt.f64 N/A

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

       7. lift-+.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 16.8

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

   14. Applied rewrites 16.8%

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

Alternative 20: 84.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 29.1%

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

9. Final simplification 29.1%

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

Alternative 21: 47.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((1.0 + x)) + sqrt((1.0 + y))) - (sqrt(x) + 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 + x)) + sqrt((1.0d0 + y))) - (sqrt(x) + 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 + x)) + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + x)) + math.sqrt((1.0 + y))) - (math.sqrt(x) + 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 + x)) + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + 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 + x)) + sqrt((1.0 + y))) - (sqrt(x) + 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 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in t around inf

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

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in z around inf

   \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
7. 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 14.9

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

8. Applied rewrites 14.9%

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

Alternative 22: 17.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + x)) + math.sqrt(y)) - (math.sqrt(x) + 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 + x)) + sqrt(y)) - Float64(sqrt(x) + 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 + x)) + sqrt(y)) - (sqrt(x) + 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 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in t around inf

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

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in z around inf

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

   1. lower--.f64 N/A

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

8. Applied rewrites 13.2%

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

9. Taylor expanded in y around inf

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

    1. lift-sqrt.f64 8.0

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

11. Applied rewrites 8.0%

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

Alternative 23: 11.4% 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 90.2%

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

3. Taylor expanded in t around inf

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

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in z around inf

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

   1. lift-sqrt.f64 2.1

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

8. Applied rewrites 2.1%

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

9. Taylor expanded in y around inf

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

    1. lift-sqrt.f64 4.0

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

11. Applied rewrites 4.0%

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

12. Taylor expanded in x around inf

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

    1. lower-+.f64 N/A

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

    2. lift-sqrt.f64 N/A

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

    3. lift-+.f64 N/A

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

    4. lower-sqrt.f64 N/A

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

    5. lower-+.f64 14.5

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

14. Applied rewrites 14.5%

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

Alternative 24: 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 90.2%

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

3. Taylor expanded in t around inf

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

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in z around inf

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

   1. lift-sqrt.f64 2.1

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

8. Applied rewrites 2.1%

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

9. Taylor expanded in y around inf

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

    1. lift-sqrt.f64 4.0

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

11. Applied rewrites 4.0%

    \[\leadsto \sqrt{z} - \sqrt{y} \]
12. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

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 = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))