Main:z from

Percentage Accurate: 91.9% → 98.3%

Time: 32.2s

Alternatives: 15

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{t + 1}\\ \mathbf{if}\;y \leq 56000000:\\ \;\;\;\;\left(\left(\left(t\_2 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + \frac{1}{t\_3 + \sqrt{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, \frac{1}{t\_2 + \sqrt{x}}\right) + t\_1\right) + \left(t\_3 - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ t 1.0))))
   (if (<= y 56000000.0)
     (+
      (+ (+ (- t_2 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) t_1)
      (/ 1.0 (+ t_3 (sqrt t))))
     (+
      (+ (fma (/ 1.0 (sqrt y)) 0.5 (/ 1.0 (+ t_2 (sqrt x)))) t_1)
      (- t_3 (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((t + 1.0));
	double tmp;
	if (y <= 56000000.0) {
		tmp = (((t_2 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + (1.0 / (t_3 + sqrt(t)));
	} else {
		tmp = (fma((1.0 / sqrt(y)), 0.5, (1.0 / (t_2 + sqrt(x)))) + t_1) + (t_3 - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (y <= 56000000.0)
		tmp = Float64(Float64(Float64(Float64(t_2 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + Float64(1.0 / Float64(t_3 + sqrt(t))));
	else
		tmp = Float64(Float64(fma(Float64(1.0 / sqrt(y)), 0.5, Float64(1.0 / Float64(t_2 + sqrt(x)))) + t_1) + Float64(t_3 - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 56000000.0], N[(N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6e7

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 97.9%

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

   if 5.6e7 < y

   1. Initial program 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 2: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 56000000:\\ \;\;\;\;\left(\left(\left(t\_3 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, \frac{1}{t\_3 + \sqrt{x}}\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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 (+ x 1.0))))
   (if (<= y 56000000.0)
     (+ (+ (+ (- t_3 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) t_1) t_2)
     (+ (+ (fma (/ 1.0 (sqrt y)) 0.5 (/ 1.0 (+ t_3 (sqrt x)))) t_1) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if (y <= 56000000.0) {
		tmp = (((t_3 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	} else {
		tmp = (fma((1.0 / sqrt(y)), 0.5, (1.0 / (t_3 + sqrt(x)))) + t_1) + t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 56000000.0)
		tmp = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_2);
	else
		tmp = Float64(Float64(fma(Float64(1.0 / sqrt(y)), 0.5, Float64(1.0 / Float64(t_3 + sqrt(x)))) + 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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 56000000.0], N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6e7

   1. Initial program 97.4%

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

   if 5.6e7 < y

   1. Initial program 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 3: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \sqrt{x + 1}\\ t_6 := \left(\left(\left(t\_5 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_4\\ \mathbf{if}\;t\_6 \leq 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{x}} + \left(\sqrt{z} - \sqrt{z}\right)\right) + t\_4\\ \mathbf{elif}\;t\_6 \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, t\_5\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\frac{1}{t\_2 + \sqrt{x}} + t\_1\right) - \sqrt{y}\\ \mathbf{elif}\;t\_6 \leq 2.999999999995:\\ \;\;\;\;\left(\left(\left(\left(t\_2 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_1 + 1\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(1 - \sqrt{z}\right)\right) + t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5 (sqrt (+ x 1.0)))
        (t_6
         (+ (+ (+ (- t_5 (sqrt x)) (- t_1 (sqrt y))) (- t_3 (sqrt z))) t_4)))
   (if (<= t_6 1e-5)
     (+ (+ (/ 0.5 (sqrt x)) (- (sqrt z) (sqrt z))) t_4)
     (if (<= t_6 1.0002)
       (+ (- (fma (/ 1.0 (sqrt y)) 0.5 t_5) (sqrt x)) t_4)
       (if (<= t_6 2.0)
         (- (+ (/ 1.0 (+ t_2 (sqrt x))) t_1) (sqrt y))
         (if (<= t_6 2.999999999995)
           (- (- (- (+ (+ t_2 t_1) t_3) (sqrt x)) (sqrt z)) (sqrt y))
           (+
            (+ (- (- (+ t_1 1.0) (sqrt x)) (sqrt y)) (- 1.0 (sqrt z)))
            t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = sqrt((x + 1.0));
	double t_6 = (((t_5 - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + t_4;
	double tmp;
	if (t_6 <= 1e-5) {
		tmp = ((0.5 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_4;
	} else if (t_6 <= 1.0002) {
		tmp = (fma((1.0 / sqrt(y)), 0.5, t_5) - sqrt(x)) + t_4;
	} else if (t_6 <= 2.0) {
		tmp = ((1.0 / (t_2 + sqrt(x))) + t_1) - sqrt(y);
	} else if (t_6 <= 2.999999999995) {
		tmp = ((((t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z)) - sqrt(y);
	} else {
		tmp = ((((t_1 + 1.0) - sqrt(x)) - sqrt(y)) + (1.0 - sqrt(z))) + t_4;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = sqrt(Float64(x + 1.0))
	t_6 = Float64(Float64(Float64(Float64(t_5 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + t_4)
	tmp = 0.0
	if (t_6 <= 1e-5)
		tmp = Float64(Float64(Float64(0.5 / sqrt(x)) + Float64(sqrt(z) - sqrt(z))) + t_4);
	elseif (t_6 <= 1.0002)
		tmp = Float64(Float64(fma(Float64(1.0 / sqrt(y)), 0.5, t_5) - sqrt(x)) + t_4);
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + t_1) - sqrt(y));
	elseif (t_6 <= 2.999999999995)
		tmp = Float64(Float64(Float64(Float64(Float64(t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z)) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 + 1.0) - sqrt(x)) - sqrt(y)) + Float64(1.0 - sqrt(z))) + t_4);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$6, 1e-5], N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$6, 1.0002], N[(N[(N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.999999999995], N[(N[(N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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))) < 1.00000000000000008e-5

   1. Initial program 6.1%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 6.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 6.1

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 84.5

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

   7. Applied rewrites 84.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

   if 1.00000000000000008e-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))) < 1.0002

   1. Initial program 94.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 1.0002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.6%

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

   4. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   6. Applied rewrites 2.1%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 97.3

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

   9. Applied rewrites 97.3%

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

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.999999999995

   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. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 91.9%

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

   if 2.999999999995 < (+.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 98.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 98.6

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

   7. Applied rewrites 98.6%

      \[\leadsto \left(\color{blue}{\left(\left(\left(\sqrt{y + 1} + 1\right) - \sqrt{x}\right) - \sqrt{y}\right)} + \left(1 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(\left(\left(t\_3 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_2\\ \mathbf{if}\;t\_4 \leq 0.9998:\\ \;\;\;\;\left(\frac{1}{t\_3 + \sqrt{x}} + t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4
         (+ (+ (+ (- t_3 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) t_1) t_2)))
   (if (<= t_4 0.9998) (+ (+ (/ 1.0 (+ t_3 (sqrt x))) t_1) t_2) t_4)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((x + 1.0));
	double t_4 = (((t_3 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	double tmp;
	if (t_4 <= 0.9998) {
		tmp = ((1.0 / (t_3 + sqrt(x))) + t_1) + t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = sqrt((x + 1.0d0))
    t_4 = (((t_3 - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1) + t_2
    if (t_4 <= 0.9998d0) then
        tmp = ((1.0d0 / (t_3 + sqrt(x))) + t_1) + t_2
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = (((t_3 - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1) + t_2;
	double tmp;
	if (t_4 <= 0.9998) {
		tmp = ((1.0 / (t_3 + Math.sqrt(x))) + t_1) + t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = math.sqrt((x + 1.0))
	t_4 = (((t_3 - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1) + t_2
	tmp = 0
	if t_4 <= 0.9998:
		tmp = ((1.0 / (t_3 + math.sqrt(x))) + t_1) + t_2
	else:
		tmp = t_4
	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 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_2)
	tmp = 0.0
	if (t_4 <= 0.9998)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(x))) + t_1) + t_2);
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = sqrt((x + 1.0));
	t_4 = (((t_3 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_2;
	tmp = 0.0;
	if (t_4 <= 0.9998)
		tmp = ((1.0 / (t_3 + sqrt(x))) + t_1) + t_2;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, 0.9998], N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], t$95$4]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 22.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 24.4%

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

   4. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 86.8

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

   6. Applied rewrites 86.8%

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

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

   1. Initial program 97.0%

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

Alternative 5: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{x + 1}\\ t_5 := \sqrt{t + 1} - \sqrt{t}\\ t_6 := \left(\left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + t\_3\right) + t\_5\\ t_7 := \sqrt{1 + x}\\ \mathbf{if}\;t\_6 \leq 1:\\ \;\;\;\;\left(\frac{1}{t\_4 + \sqrt{x}} + t\_3\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\frac{1}{t\_7 + \sqrt{x}} + t\_1\right) - \sqrt{y}\\ \mathbf{elif}\;t\_6 \leq 2.999999999995:\\ \;\;\;\;\left(\left(\left(\left(t\_7 + t\_1\right) + t\_2\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_1 + 1\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(1 - \sqrt{z}\right)\right) + t\_5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_6 (+ (+ (+ (- t_4 (sqrt x)) (- t_1 (sqrt y))) t_3) t_5))
        (t_7 (sqrt (+ 1.0 x))))
   (if (<= t_6 1.0)
     (+ (+ (/ 1.0 (+ t_4 (sqrt x))) t_3) t_5)
     (if (<= t_6 2.0)
       (- (+ (/ 1.0 (+ t_7 (sqrt x))) t_1) (sqrt y))
       (if (<= t_6 2.999999999995)
         (- (- (- (+ (+ t_7 t_1) t_2) (sqrt x)) (sqrt z)) (sqrt y))
         (+
          (+ (- (- (+ t_1 1.0) (sqrt x)) (sqrt y)) (- 1.0 (sqrt z)))
          t_5))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((x + 1.0));
	double t_5 = sqrt((t + 1.0)) - sqrt(t);
	double t_6 = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + t_3) + t_5;
	double t_7 = sqrt((1.0 + x));
	double tmp;
	if (t_6 <= 1.0) {
		tmp = ((1.0 / (t_4 + sqrt(x))) + t_3) + t_5;
	} else if (t_6 <= 2.0) {
		tmp = ((1.0 / (t_7 + sqrt(x))) + t_1) - sqrt(y);
	} else if (t_6 <= 2.999999999995) {
		tmp = ((((t_7 + t_1) + t_2) - sqrt(x)) - sqrt(z)) - sqrt(y);
	} else {
		tmp = ((((t_1 + 1.0) - sqrt(x)) - sqrt(y)) + (1.0 - sqrt(z))) + t_5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((x + 1.0d0))
    t_5 = sqrt((t + 1.0d0)) - sqrt(t)
    t_6 = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + t_3) + t_5
    t_7 = sqrt((1.0d0 + x))
    if (t_6 <= 1.0d0) then
        tmp = ((1.0d0 / (t_4 + sqrt(x))) + t_3) + t_5
    else if (t_6 <= 2.0d0) then
        tmp = ((1.0d0 / (t_7 + sqrt(x))) + t_1) - sqrt(y)
    else if (t_6 <= 2.999999999995d0) then
        tmp = ((((t_7 + t_1) + t_2) - sqrt(x)) - sqrt(z)) - sqrt(y)
    else
        tmp = ((((t_1 + 1.0d0) - sqrt(x)) - sqrt(y)) + (1.0d0 - sqrt(z))) + t_5
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((x + 1.0));
	double t_5 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_6 = (((t_4 - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + t_3) + t_5;
	double t_7 = Math.sqrt((1.0 + x));
	double tmp;
	if (t_6 <= 1.0) {
		tmp = ((1.0 / (t_4 + Math.sqrt(x))) + t_3) + t_5;
	} else if (t_6 <= 2.0) {
		tmp = ((1.0 / (t_7 + Math.sqrt(x))) + t_1) - Math.sqrt(y);
	} else if (t_6 <= 2.999999999995) {
		tmp = ((((t_7 + t_1) + t_2) - Math.sqrt(x)) - Math.sqrt(z)) - Math.sqrt(y);
	} else {
		tmp = ((((t_1 + 1.0) - Math.sqrt(x)) - Math.sqrt(y)) + (1.0 - Math.sqrt(z))) + t_5;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((x + 1.0))
	t_5 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_6 = (((t_4 - math.sqrt(x)) + (t_1 - math.sqrt(y))) + t_3) + t_5
	t_7 = math.sqrt((1.0 + x))
	tmp = 0
	if t_6 <= 1.0:
		tmp = ((1.0 / (t_4 + math.sqrt(x))) + t_3) + t_5
	elif t_6 <= 2.0:
		tmp = ((1.0 / (t_7 + math.sqrt(x))) + t_1) - math.sqrt(y)
	elif t_6 <= 2.999999999995:
		tmp = ((((t_7 + t_1) + t_2) - math.sqrt(x)) - math.sqrt(z)) - math.sqrt(y)
	else:
		tmp = ((((t_1 + 1.0) - math.sqrt(x)) - math.sqrt(y)) + (1.0 - math.sqrt(z))) + t_5
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_6 = Float64(Float64(Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y))) + t_3) + t_5)
	t_7 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_6 <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(x))) + t_3) + t_5);
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_7 + sqrt(x))) + t_1) - sqrt(y));
	elseif (t_6 <= 2.999999999995)
		tmp = Float64(Float64(Float64(Float64(Float64(t_7 + t_1) + t_2) - sqrt(x)) - sqrt(z)) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 + 1.0) - sqrt(x)) - sqrt(y)) + Float64(1.0 - sqrt(z))) + t_5);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((x + 1.0));
	t_5 = sqrt((t + 1.0)) - sqrt(t);
	t_6 = (((t_4 - sqrt(x)) + (t_1 - sqrt(y))) + t_3) + t_5;
	t_7 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_6 <= 1.0)
		tmp = ((1.0 / (t_4 + sqrt(x))) + t_3) + t_5;
	elseif (t_6 <= 2.0)
		tmp = ((1.0 / (t_7 + sqrt(x))) + t_1) - sqrt(y);
	elseif (t_6 <= 2.999999999995)
		tmp = ((((t_7 + t_1) + t_2) - sqrt(x)) - sqrt(z)) - sqrt(y);
	else
		tmp = ((((t_1 + 1.0) - sqrt(x)) - sqrt(y)) + (1.0 - sqrt(z))) + t_5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 1.0], N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(1.0 / N[(t$95$7 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.999999999995], N[(N[(N[(N[(N[(t$95$7 + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 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 79.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 79.4%

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

   4. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 94.6

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

   6. Applied rewrites 94.6%

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

   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

   1. Initial program 96.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   6. Applied rewrites 2.1%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 96.3

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

   9. Applied rewrites 96.3%

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

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.999999999995

   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. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 91.9%

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

   if 2.999999999995 < (+.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 98.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 6: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;\left(\left(\left(t\_3 - \sqrt{x}\right) + t\_4\right) + t\_1\right) + t\_2 \leq 0.9999998:\\ \;\;\;\;\left(\frac{1}{t\_3 + \sqrt{x}} + t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_4\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_4 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= (+ (+ (+ (- t_3 (sqrt x)) t_4) t_1) t_2) 0.9999998)
     (+ (+ (/ 1.0 (+ t_3 (sqrt x))) t_1) t_2)
     (+ (+ (+ (- 1.0 (sqrt x)) t_4) 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((x + 1.0));
	double t_4 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (((((t_3 - sqrt(x)) + t_4) + t_1) + t_2) <= 0.9999998) {
		tmp = ((1.0 / (t_3 + sqrt(x))) + t_1) + t_2;
	} else {
		tmp = (((1.0 - sqrt(x)) + t_4) + t_1) + t_2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = sqrt((x + 1.0d0))
    t_4 = sqrt((y + 1.0d0)) - sqrt(y)
    if (((((t_3 - sqrt(x)) + t_4) + t_1) + t_2) <= 0.9999998d0) then
        tmp = ((1.0d0 / (t_3 + sqrt(x))) + t_1) + t_2
    else
        tmp = (((1.0d0 - sqrt(x)) + t_4) + t_1) + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if (((((t_3 - Math.sqrt(x)) + t_4) + t_1) + t_2) <= 0.9999998) {
		tmp = ((1.0 / (t_3 + Math.sqrt(x))) + t_1) + t_2;
	} else {
		tmp = (((1.0 - Math.sqrt(x)) + t_4) + t_1) + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = math.sqrt((x + 1.0))
	t_4 = math.sqrt((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if ((((t_3 - math.sqrt(x)) + t_4) + t_1) + t_2) <= 0.9999998:
		tmp = ((1.0 / (t_3 + math.sqrt(x))) + t_1) + t_2
	else:
		tmp = (((1.0 - math.sqrt(x)) + t_4) + 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 = sqrt(Float64(x + 1.0))
	t_4 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t_3 - sqrt(x)) + t_4) + t_1) + t_2) <= 0.9999998)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(x))) + t_1) + t_2);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_4) + t_1) + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = sqrt((x + 1.0));
	t_4 = sqrt((y + 1.0)) - sqrt(y);
	tmp = 0.0;
	if (((((t_3 - sqrt(x)) + t_4) + t_1) + t_2) <= 0.9999998)
		tmp = ((1.0 / (t_3 + sqrt(x))) + t_1) + t_2;
	else
		tmp = (((1.0 - sqrt(x)) + t_4) + t_1) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], 0.9999998], N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $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))) < 0.999999799999999994

   1. Initial program 27.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 29.4%

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

   4. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 87.5

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

   6. Applied rewrites 87.5%

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

   if 0.999999799999999994 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 96.9%

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

Alternative 7: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \sqrt{x + 1}\\ t_6 := \left(\left(\left(t\_5 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_4\\ \mathbf{if}\;t\_6 \leq 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{x}} + \left(\sqrt{z} - \sqrt{z}\right)\right) + t\_4\\ \mathbf{elif}\;t\_6 \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, t\_5\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\frac{1}{t\_2 + \sqrt{x}} + t\_1\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_2 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \sqrt{z}\right) - \sqrt{y}\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5 (sqrt (+ x 1.0)))
        (t_6
         (+ (+ (+ (- t_5 (sqrt x)) (- t_1 (sqrt y))) (- t_3 (sqrt z))) t_4)))
   (if (<= t_6 1e-5)
     (+ (+ (/ 0.5 (sqrt x)) (- (sqrt z) (sqrt z))) t_4)
     (if (<= t_6 1.0002)
       (+ (- (fma (/ 1.0 (sqrt y)) 0.5 t_5) (sqrt x)) t_4)
       (if (<= t_6 2.0)
         (- (+ (/ 1.0 (+ t_2 (sqrt x))) t_1) (sqrt y))
         (- (- (- (+ (+ t_2 t_1) t_3) (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((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = sqrt((x + 1.0));
	double t_6 = (((t_5 - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + t_4;
	double tmp;
	if (t_6 <= 1e-5) {
		tmp = ((0.5 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_4;
	} else if (t_6 <= 1.0002) {
		tmp = (fma((1.0 / sqrt(y)), 0.5, t_5) - sqrt(x)) + t_4;
	} else if (t_6 <= 2.0) {
		tmp = ((1.0 / (t_2 + sqrt(x))) + t_1) - sqrt(y);
	} else {
		tmp = ((((t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z)) - sqrt(y);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = sqrt(Float64(x + 1.0))
	t_6 = Float64(Float64(Float64(Float64(t_5 - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + t_4)
	tmp = 0.0
	if (t_6 <= 1e-5)
		tmp = Float64(Float64(Float64(0.5 / sqrt(x)) + Float64(sqrt(z) - sqrt(z))) + t_4);
	elseif (t_6 <= 1.0002)
		tmp = Float64(Float64(fma(Float64(1.0 / sqrt(y)), 0.5, t_5) - sqrt(x)) + t_4);
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(x))) + t_1) - sqrt(y));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_2 + t_1) + t_3) - sqrt(x)) - sqrt(z)) - sqrt(y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$6, 1e-5], N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$6, 1.0002], N[(N[(N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 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))) < 1.00000000000000008e-5

   1. Initial program 6.1%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 6.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 6.1

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 84.5

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

   7. Applied rewrites 84.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

   if 1.00000000000000008e-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))) < 1.0002

   1. Initial program 94.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 1.0002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.6%

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

   4. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   6. Applied rewrites 2.1%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 97.3

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

   9. Applied rewrites 97.3%

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

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

   1. Initial program 97.2%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 81.2%

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

Alternative 8: 70.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1}\\ t_4 := \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\\ \mathbf{if}\;t\_4 \leq 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{x}} + \left(\sqrt{z} - \sqrt{z}\right)\right) + t\_2\\ \mathbf{elif}\;t\_4 \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, t\_1\right) - \sqrt{x}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{1 + x} + t\_3\right) - \sqrt{y}\right) - \sqrt{x}\right) + t\_2\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4
         (+
          (+ (- t_1 (sqrt x)) (- t_3 (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z)))))
   (if (<= t_4 1e-5)
     (+ (+ (/ 0.5 (sqrt x)) (- (sqrt z) (sqrt z))) t_2)
     (if (<= t_4 1.0002)
       (+ (- (fma (/ 1.0 (sqrt y)) 0.5 t_1) (sqrt x)) t_2)
       (+ (- (- (+ (sqrt (+ 1.0 x)) t_3) (sqrt y)) (sqrt x)) t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((y + 1.0));
	double t_4 = ((t_1 - sqrt(x)) + (t_3 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	double tmp;
	if (t_4 <= 1e-5) {
		tmp = ((0.5 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_2;
	} else if (t_4 <= 1.0002) {
		tmp = (fma((1.0 / sqrt(y)), 0.5, t_1) - sqrt(x)) + t_2;
	} else {
		tmp = (((sqrt((1.0 + x)) + t_3) - sqrt(y)) - sqrt(x)) + t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(t_3 - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1e-5)
		tmp = Float64(Float64(Float64(0.5 / sqrt(x)) + Float64(sqrt(z) - sqrt(z))) + t_2);
	elseif (t_4 <= 1.0002)
		tmp = Float64(Float64(fma(Float64(1.0 / sqrt(y)), 0.5, t_1) - sqrt(x)) + t_2);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + t_3) - sqrt(y)) - sqrt(x)) + 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[(x + 1.0), $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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-5], N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1.0002], N[(N[(N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $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))) < 1.00000000000000008e-5

   1. Initial program 6.1%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 6.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 6.1

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 84.5

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

   7. Applied rewrites 84.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

   if 1.00000000000000008e-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))) < 1.0002

   1. Initial program 94.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 1.0002 < (+.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 97.4%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 59.9

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

   4. Applied rewrites 59.9%

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

Alternative 9: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1}\\ t_4 := \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\\ \mathbf{if}\;t\_4 \leq 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{x}} + \left(\sqrt{z} - \sqrt{z}\right)\right) + t\_2\\ \mathbf{elif}\;t\_4 \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{\sqrt{y}}, 0.5, t\_1\right) - \sqrt{x}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + t\_3\right) - \sqrt{y}\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4
         (+
          (+ (- t_1 (sqrt x)) (- t_3 (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z)))))
   (if (<= t_4 1e-5)
     (+ (+ (/ 0.5 (sqrt x)) (- (sqrt z) (sqrt z))) t_2)
     (if (<= t_4 1.0002)
       (+ (- (fma (/ 1.0 (sqrt y)) 0.5 t_1) (sqrt x)) t_2)
       (- (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) t_3) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((y + 1.0));
	double t_4 = ((t_1 - sqrt(x)) + (t_3 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	double tmp;
	if (t_4 <= 1e-5) {
		tmp = ((0.5 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_2;
	} else if (t_4 <= 1.0002) {
		tmp = (fma((1.0 / sqrt(y)), 0.5, t_1) - sqrt(x)) + t_2;
	} else {
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + t_3) - sqrt(y);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(Float64(Float64(t_1 - sqrt(x)) + Float64(t_3 - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1e-5)
		tmp = Float64(Float64(Float64(0.5 / sqrt(x)) + Float64(sqrt(z) - sqrt(z))) + t_2);
	elseif (t_4 <= 1.0002)
		tmp = Float64(Float64(fma(Float64(1.0 / sqrt(y)), 0.5, t_1) - sqrt(x)) + t_2);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + t_3) - sqrt(y));
	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[(x + 1.0), $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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-5], N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1.0002], N[(N[(N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[y], $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))) < 1.00000000000000008e-5

   1. Initial program 6.1%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 6.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 6.1

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 84.5

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

   7. Applied rewrites 84.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

   if 1.00000000000000008e-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))) < 1.0002

   1. Initial program 94.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 17.5

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 1.0002 < (+.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 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.4%

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

   4. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   6. Applied rewrites 4.4%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 59.8

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

   9. Applied rewrites 59.8%

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

Alternative 10: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1} - \sqrt{x}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(t\_2 + \left(t\_1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{x}} + \left(\sqrt{z} - \sqrt{z}\right)\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;t\_2 + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + t\_1\right) - \sqrt{y}\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+ (+ (+ t_2 (- t_1 (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) t_3)))
   (if (<= t_4 1e-5)
     (+ (+ (/ 0.5 (sqrt x)) (- (sqrt z) (sqrt z))) t_3)
     (if (<= t_4 1.0)
       (+ t_2 t_3)
       (- (+ (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))) t_1) (sqrt y))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0)) - sqrt(x);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = ((t_2 + (t_1 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_3;
	double tmp;
	if (t_4 <= 1e-5) {
		tmp = ((0.5 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_3;
	} else if (t_4 <= 1.0) {
		tmp = t_2 + t_3;
	} else {
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + t_1) - 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((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0)) - sqrt(x)
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = ((t_2 + (t_1 - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + t_3
    if (t_4 <= 1d-5) then
        tmp = ((0.5d0 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_3
    else if (t_4 <= 1.0d0) then
        tmp = t_2 + t_3
    else
        tmp = ((1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))) + t_1) - 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((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = ((t_2 + (t_1 - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + t_3;
	double tmp;
	if (t_4 <= 1e-5) {
		tmp = ((0.5 / Math.sqrt(x)) + (Math.sqrt(z) - Math.sqrt(z))) + t_3;
	} else if (t_4 <= 1.0) {
		tmp = t_2 + t_3;
	} else {
		tmp = ((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + t_1) - 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((y + 1.0))
	t_2 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = ((t_2 + (t_1 - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + t_3
	tmp = 0
	if t_4 <= 1e-5:
		tmp = ((0.5 / math.sqrt(x)) + (math.sqrt(z) - math.sqrt(z))) + t_3
	elif t_4 <= 1.0:
		tmp = t_2 + t_3
	else:
		tmp = ((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + t_1) - 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(y + 1.0))
	t_2 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(t_2 + Float64(t_1 - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_3)
	tmp = 0.0
	if (t_4 <= 1e-5)
		tmp = Float64(Float64(Float64(0.5 / sqrt(x)) + Float64(sqrt(z) - sqrt(z))) + t_3);
	elseif (t_4 <= 1.0)
		tmp = Float64(t_2 + t_3);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + t_1) - 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((y + 1.0));
	t_2 = sqrt((x + 1.0)) - sqrt(x);
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = ((t_2 + (t_1 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_3;
	tmp = 0.0;
	if (t_4 <= 1e-5)
		tmp = ((0.5 / sqrt(x)) + (sqrt(z) - sqrt(z))) + t_3;
	elseif (t_4 <= 1.0)
		tmp = t_2 + t_3;
	else
		tmp = ((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + t_1) - 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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-5], N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(t$95$2 + t$95$3), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $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))) < 1.00000000000000008e-5

   1. Initial program 6.1%

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

   2. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 6.1

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 6.1

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 84.5

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

   7. Applied rewrites 84.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.5%

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

   if 1.00000000000000008e-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

   1. Initial program 96.4%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 13.5

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in y around inf

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

      1. lift-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 96.3

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

   7. Applied rewrites 96.3%

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

   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)))

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 96.9%

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

   4. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   6. Applied rewrites 4.4%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 60.1

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

   9. Applied rewrites 60.1%

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

Alternative 11: 65.4% accurate, 0.7× speedup

Math

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

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

C

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

   1. Initial program 79.0%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 11.4

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

   4. Applied rewrites 11.4%

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

   5. Taylor expanded in y around inf

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

      1. lift-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 78.9

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

   7. Applied rewrites 78.9%

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

   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)))

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 96.9%

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

   4. Taylor expanded in z around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   6. Applied rewrites 4.4%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 60.1

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

   9. Applied rewrites 60.1%

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

Alternative 12: 36.1% accurate, 2.1× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (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 (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((t + 1.0)) - 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 = (sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((t + 1.0d0)) - 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 (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 46.5

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

4. Applied rewrites 46.5%

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

5. Taylor expanded in y around inf

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

   1. lift-sqrt.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 36.1

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

7. Applied rewrites 36.1%

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

Alternative 13: 14.0% accurate, 2.1× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- (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 (sqrt((y + 1.0)) - sqrt(y)) + (sqrt((t + 1.0)) - 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 = (sqrt((y + 1.0d0)) - sqrt(y)) + (sqrt((t + 1.0d0)) - 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 (Math.sqrt((y + 1.0)) - Math.sqrt(y)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 46.5

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

4. Applied rewrites 46.5%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 14.0

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

7. Applied rewrites 14.0%

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

Alternative 14: 6.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt(x) - sqrt(x)) + ((1.0d0 / sqrt(t)) * 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 46.5

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

4. Applied rewrites 46.5%

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

5. Taylor expanded in x around inf

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

   1. lift-sqrt.f64 4.2

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

7. Applied rewrites 4.2%

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

8. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. lift-sqrt.f64 6.2

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

10. Applied rewrites 6.2%

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

Alternative 15: 3.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. flip-- N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 92.0%

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

4. Taylor expanded in z around inf

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

   1. associate--r+ N/A

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

   2. lower--.f64 N/A

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

6. Applied rewrites 3.6%

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

7. Taylor expanded in y around inf

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

   1. lift-sqrt.f64 3.1

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

9. Applied rewrites 3.1%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64
  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))