Main:z from

Percentage Accurate: 91.8% → 99.5%

Time: 13.7s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := t\_2 + t\_4\\ \mathbf{if}\;t\_5 \leq 0.001:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + t\_1\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.001:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \frac{1}{t\_3 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_1))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (+ t_2 t_4)))
   (if (<= t_5 0.001)
     (+
      (+
       (+
        (/
         (fma
          -0.125
          (/ 1.0 (sqrt x))
          (fma 0.0625 (sqrt (/ 1.0 (* (* x x) x))) (* 0.5 (sqrt x))))
         x)
        (* (/ 1.0 (sqrt y)) 0.5))
       t_1)
      t_4)
     (if (<= t_5 2.001)
       (-
        (+
         (sqrt (+ 1.0 x))
         (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
        (sqrt x))
       (+ t_2 (/ 1.0 (+ 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)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 - sqrt(t);
	double t_5 = t_2 + t_4;
	double tmp;
	if (t_5 <= 0.001) {
		tmp = (((fma(-0.125, (1.0 / sqrt(x)), fma(0.0625, sqrt((1.0 / ((x * x) * x))), (0.5 * sqrt(x)))) / x) + ((1.0 / sqrt(y)) * 0.5)) + t_1) + t_4;
	} else if (t_5 <= 2.001) {
		tmp = (sqrt((1.0 + x)) + fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(y) + sqrt((1.0 + y)))))) - sqrt(x);
	} else {
		tmp = t_2 + (1.0 / (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 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1)
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = Float64(t_2 + t_4)
	tmp = 0.0
	if (t_5 <= 0.001)
		tmp = Float64(Float64(Float64(Float64(fma(-0.125, Float64(1.0 / sqrt(x)), fma(0.0625, sqrt(Float64(1.0 / Float64(Float64(x * x) * x))), Float64(0.5 * sqrt(x)))) / x) + Float64(Float64(1.0 / sqrt(y)) * 0.5)) + t_1) + t_4);
	elseif (t_5 <= 2.001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))) - sqrt(x));
	else
		tmp = Float64(t_2 + Float64(1.0 / 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[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 0.001], N[(N[(N[(N[(N[(-0.125 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[Sqrt[N[(1.0 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.001], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(1.0 / N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1e-3

   1. Initial program 9.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{\color{blue}{x}} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. sqrt-div N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{\sqrt{1}}{\sqrt{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. unpow3 N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lift-sqrt.f64 98.1

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \sqrt{x}\right)\right)}{x} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 98.1%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(-0.125, \frac{1}{\sqrt{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \sqrt{x}\right)\right)}{x}} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1e-3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00099999999999989

   1. Initial program 95.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 96.1%

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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 11.0%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 99.4%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. 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. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 98.8%

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

   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 99.8%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := t\_2 + t\_4\\ \mathbf{if}\;t\_5 \leq 0.001:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{\sqrt{x}}, 0.5 \cdot \sqrt{x}\right)}{x} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + t\_1\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.001:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \frac{1}{t\_3 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_1))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (+ t_2 t_4)))
   (if (<= t_5 0.001)
     (+
      (+
       (+
        (/ (fma -0.125 (/ 1.0 (sqrt x)) (* 0.5 (sqrt x))) x)
        (* (/ 1.0 (sqrt y)) 0.5))
       t_1)
      t_4)
     (if (<= t_5 2.001)
       (-
        (+
         (sqrt (+ 1.0 x))
         (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
        (sqrt x))
       (+ t_2 (/ 1.0 (+ 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)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 - sqrt(t);
	double t_5 = t_2 + t_4;
	double tmp;
	if (t_5 <= 0.001) {
		tmp = (((fma(-0.125, (1.0 / sqrt(x)), (0.5 * sqrt(x))) / x) + ((1.0 / sqrt(y)) * 0.5)) + t_1) + t_4;
	} else if (t_5 <= 2.001) {
		tmp = (sqrt((1.0 + x)) + fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(y) + sqrt((1.0 + y)))))) - sqrt(x);
	} else {
		tmp = t_2 + (1.0 / (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 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1)
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = Float64(t_2 + t_4)
	tmp = 0.0
	if (t_5 <= 0.001)
		tmp = Float64(Float64(Float64(Float64(fma(-0.125, Float64(1.0 / sqrt(x)), Float64(0.5 * sqrt(x))) / x) + Float64(Float64(1.0 / sqrt(y)) * 0.5)) + t_1) + t_4);
	elseif (t_5 <= 2.001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))) - sqrt(x));
	else
		tmp = Float64(t_2 + Float64(1.0 / 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[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 0.001], N[(N[(N[(N[(N[(-0.125 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.001], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(1.0 / N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1e-3

   1. Initial program 9.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 97.8

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

   7. Applied rewrites 97.8%

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

   if 1e-3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00099999999999989

   1. Initial program 95.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 96.1%

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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 11.0%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 99.4%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. 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. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 98.8%

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

   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 99.8%

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

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := t\_2 + t\_4\\ \mathbf{if}\;t\_5 \leq 10^{-5}:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + t\_1\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.001:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \frac{1}{t\_3 + \sqrt{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2
         (+
          (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          t_1))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (+ t_2 t_4)))
   (if (<= t_5 1e-5)
     (+ (+ (+ (* 0.5 (/ 1.0 (sqrt x))) (* (/ 1.0 (sqrt y)) 0.5)) t_1) t_4)
     (if (<= t_5 2.001)
       (-
        (+
         (sqrt (+ 1.0 x))
         (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
        (sqrt x))
       (+ t_2 (/ 1.0 (+ 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)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 - sqrt(t);
	double t_5 = t_2 + t_4;
	double tmp;
	if (t_5 <= 1e-5) {
		tmp = (((0.5 * (1.0 / sqrt(x))) + ((1.0 / sqrt(y)) * 0.5)) + t_1) + t_4;
	} else if (t_5 <= 2.001) {
		tmp = (sqrt((1.0 + x)) + fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(y) + sqrt((1.0 + y)))))) - sqrt(x);
	} else {
		tmp = t_2 + (1.0 / (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 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1)
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = Float64(t_2 + t_4)
	tmp = 0.0
	if (t_5 <= 1e-5)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + Float64(Float64(1.0 / sqrt(y)) * 0.5)) + t_1) + t_4);
	elseif (t_5 <= 2.001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))) - sqrt(x));
	else
		tmp = Float64(t_2 + Float64(1.0 / 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[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 1e-5], N[(N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.001], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(1.0 / N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00000000000000008e-5

   1. Initial program 6.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 12.0

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 96.9

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

   7. Applied rewrites 96.9%

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

   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))) < 2.00099999999999989

   1. Initial program 95.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 95.8%

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

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 11.0%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 99.2%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. 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. flip-- N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 98.8%

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

   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 99.8%

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

Alternative 4: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ t_5 := \sqrt{y + 1} - \sqrt{y}\\ t_6 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_5\right) + t\_2\right) + t\_3\\ \mathbf{if}\;t\_6 \leq 10^{-5}:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + t\_2\right) + t\_3\\ \mathbf{elif}\;t\_6 \leq 2.001:\\ \;\;\;\;\left(t\_4 + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_4 - \left(\sqrt{x} - t\_5\right)\right) + t\_1\right) - \sqrt{z}\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_6 (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_5) t_2) t_3)))
   (if (<= t_6 1e-5)
     (+ (+ (+ (* 0.5 (/ 1.0 (sqrt x))) (* (/ 1.0 (sqrt y)) 0.5)) t_2) t_3)
     (if (<= t_6 2.001)
       (-
        (+
         t_4
         (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
        (sqrt x))
       (+ (- (+ (- t_4 (- (sqrt x) t_5)) t_1) (sqrt z)) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((1.0 + x));
	double t_5 = sqrt((y + 1.0)) - sqrt(y);
	double t_6 = (((sqrt((x + 1.0)) - sqrt(x)) + t_5) + t_2) + t_3;
	double tmp;
	if (t_6 <= 1e-5) {
		tmp = (((0.5 * (1.0 / sqrt(x))) + ((1.0 / sqrt(y)) * 0.5)) + t_2) + t_3;
	} else if (t_6 <= 2.001) {
		tmp = (t_4 + fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(y) + sqrt((1.0 + y)))))) - sqrt(x);
	} else {
		tmp = (((t_4 - (sqrt(x) - t_5)) + t_1) - sqrt(z)) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_6 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_5) + t_2) + t_3)
	tmp = 0.0
	if (t_6 <= 1e-5)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + Float64(Float64(1.0 / sqrt(y)) * 0.5)) + t_2) + t_3);
	elseif (t_6 <= 2.001)
		tmp = Float64(Float64(t_4 + fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(Float64(t_4 - Float64(sqrt(x) - t_5)) + t_1) - sqrt(z)) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$6, 1e-5], N[(N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$6, 2.001], N[(N[(t$95$4 + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 - N[(N[Sqrt[x], $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00000000000000008e-5

   1. Initial program 6.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 12.0

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 96.9

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

   7. Applied rewrites 96.9%

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

   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))) < 2.00099999999999989

   1. Initial program 95.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 95.8%

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

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 11.0%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 99.2%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.6%

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

   3. Applied rewrites 98.6%

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

Alternative 5: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + t\_2\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 10^{-5}:\\ \;\;\;\;\left(\left(0.5 \cdot \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} \cdot 0.5\right) + t\_2\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 2.001:\\ \;\;\;\;\left(\sqrt{1 + x} + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_1 + 1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_2\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y))) t_2) t_3)))
   (if (<= t_4 1e-5)
     (+ (+ (+ (* 0.5 (/ 1.0 (sqrt x))) (* (/ 1.0 (sqrt y)) 0.5)) t_2) t_3)
     (if (<= t_4 2.001)
       (-
        (+
         (sqrt (+ 1.0 x))
         (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
        (sqrt x))
       (+ (+ (- (- (+ t_1 1.0) (sqrt x)) (sqrt y)) t_2) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0)) - sqrt(z);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + t_2) + t_3;
	double tmp;
	if (t_4 <= 1e-5) {
		tmp = (((0.5 * (1.0 / sqrt(x))) + ((1.0 / sqrt(y)) * 0.5)) + t_2) + t_3;
	} else if (t_4 <= 2.001) {
		tmp = (sqrt((1.0 + x)) + fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(y) + sqrt((1.0 + y)))))) - sqrt(x);
	} else {
		tmp = ((((t_1 + 1.0) - sqrt(x)) - sqrt(y)) + t_2) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + t_2) + t_3)
	tmp = 0.0
	if (t_4 <= 1e-5)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + Float64(Float64(1.0 / sqrt(y)) * 0.5)) + t_2) + t_3);
	elseif (t_4 <= 2.001)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) + fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))) - sqrt(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 + 1.0) - sqrt(x)) - sqrt(y)) + t_2) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-5], N[(N[(N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 2.001], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00000000000000008e-5

   1. Initial program 6.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 12.0

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 96.9

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

   7. Applied rewrites 96.9%

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

   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))) < 2.00099999999999989

   1. Initial program 95.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 95.8%

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

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 11.0%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 99.2%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 6: 94.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.99999999999999991e-6

   1. Initial program 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 8.9%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 91.9%

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

   if 1.99999999999999991e-6 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 98.0%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 7: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\\ t_5 := \left(t\_4 + \left(t\_3 - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_5 \leq 2.001:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 3.5:\\ \;\;\;\;\left(\left(\left(t\_2 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 + \left(1 - \sqrt{z}\right)\right) + \left(1 - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y))))
        (t_5 (+ (+ t_4 (- t_3 (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_5 2.001)
     (-
      (+ t_2 (fma 0.5 (/ 1.0 (sqrt z)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))))
      (sqrt x))
     (if (<= t_5 3.5)
       (- (- (+ (+ t_2 t_1) t_3) (sqrt x)) (+ (sqrt z) (sqrt y)))
       (+ (+ t_4 (- 1.0 (sqrt z))) (- 1.0 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = (sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y));
	double t_5 = (t_4 + (t_3 - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_5 <= 2.001) {
		tmp = (t_2 + fma(0.5, (1.0 / sqrt(z)), (1.0 / (sqrt(y) + sqrt((1.0 + y)))))) - sqrt(x);
	} else if (t_5 <= 3.5) {
		tmp = (((t_2 + t_1) + t_3) - sqrt(x)) - (sqrt(z) + sqrt(y));
	} else {
		tmp = (t_4 + (1.0 - sqrt(z))) + (1.0 - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y)))
	t_5 = Float64(Float64(t_4 + Float64(t_3 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_5 <= 2.001)
		tmp = Float64(Float64(t_2 + fma(0.5, Float64(1.0 / sqrt(z)), Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y)))))) - sqrt(x));
	elseif (t_5 <= 3.5)
		tmp = Float64(Float64(Float64(Float64(t_2 + t_1) + t_3) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(Float64(t_4 + Float64(1.0 - sqrt(z))) + Float64(1.0 - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 2.001], N[(N[(t$95$2 + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.5], N[(N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00099999999999989

   1. Initial program 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 88.7%

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

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 10.4%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 91.7%

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

   if 2.00099999999999989 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 98.3%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

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

   if 3.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 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 99.3%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 91.3%

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

Alternative 8: 88.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 38000000:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + z} + t\_1\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \mathsf{fma}\left(0.5, \frac{1}{\sqrt{z}}, t\_1\right)\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= z 38000000.0)
     (- (+ t_2 (+ (sqrt (+ 1.0 z)) t_1)) (sqrt z))
     (- (+ t_2 (fma 0.5 (/ 1.0 (sqrt z)) t_1)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (z <= 38000000.0) {
		tmp = (t_2 + (sqrt((1.0 + z)) + t_1)) - sqrt(z);
	} else {
		tmp = (t_2 + fma(0.5, (1.0 / sqrt(z)), t_1)) - sqrt(x);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (z <= 38000000.0)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + z)) + t_1)) - sqrt(z));
	else
		tmp = Float64(Float64(t_2 + fma(0.5, Float64(1.0 / sqrt(z)), t_1)) - sqrt(x));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 38000000.0], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(0.5 * N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 3.8e7

   1. Initial program 98.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 98.4%

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in z around inf

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

       1. lift-sqrt.f64 81.2

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

   11. Applied rewrites 81.2%

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

   if 3.8e7 < z

   1. Initial program 88.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 88.7%

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

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 92.0%

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

Alternative 9: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0:\\ \;\;\;\;\left(t\_2 + t\_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + z} + t\_1\right)\right) - \sqrt{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.0)
     (- (+ t_2 t_1) (sqrt x))
     (- (+ t_2 (+ (sqrt (+ 1.0 z)) t_1)) (sqrt z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.0) {
		tmp = (t_2 + t_1) - sqrt(x);
	} else {
		tmp = (t_2 + (sqrt((1.0 + z)) + t_1)) - sqrt(z);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))
    t_2 = sqrt((1.0d0 + x))
    if ((sqrt((z + 1.0d0)) - sqrt(z)) <= 0.0d0) then
        tmp = (t_2 + t_1) - sqrt(x)
    else
        tmp = (t_2 + (sqrt((1.0d0 + z)) + t_1)) - sqrt(z)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if ((Math.sqrt((z + 1.0)) - Math.sqrt(z)) <= 0.0) {
		tmp = (t_2 + t_1) - Math.sqrt(x);
	} else {
		tmp = (t_2 + (Math.sqrt((1.0 + z)) + t_1)) - Math.sqrt(z);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(t$95$2 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0

   1. Initial program 88.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 89.1%

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

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 7.7%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 90.8

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

   11. Applied rewrites 90.8%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 97.0%

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

   5. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 80.8%

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

   9. Taylor expanded in z around inf

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

       1. lift-sqrt.f64 80.8

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

   11. Applied rewrites 80.8%

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

Alternative 10: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{y} + \sqrt{1 + y}}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 0.88:\\ \;\;\;\;\left(t\_2 + \left(\sqrt{1 + z} + t\_1\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + t\_1\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (t_2 (sqrt (+ 1.0 x))))
   (if (<= z 0.88)
     (- (+ t_2 (+ (sqrt (+ 1.0 z)) t_1)) (sqrt x))
     (- (+ t_2 t_1) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	double t_2 = sqrt((1.0 + x));
	double tmp;
	if (z <= 0.88) {
		tmp = (t_2 + (sqrt((1.0 + z)) + t_1)) - sqrt(x);
	} else {
		tmp = (t_2 + t_1) - sqrt(x);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 / (sqrt(y) + sqrt((1.0d0 + y)))
    t_2 = sqrt((1.0d0 + x))
    if (z <= 0.88d0) then
        tmp = (t_2 + (sqrt((1.0d0 + z)) + t_1)) - sqrt(x)
    else
        tmp = (t_2 + t_1) - sqrt(x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / (Math.sqrt(y) + Math.sqrt((1.0 + y)));
	double t_2 = Math.sqrt((1.0 + x));
	double tmp;
	if (z <= 0.88) {
		tmp = (t_2 + (Math.sqrt((1.0 + z)) + t_1)) - Math.sqrt(x);
	} else {
		tmp = (t_2 + t_1) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 / (math.sqrt(y) + math.sqrt((1.0 + y)))
	t_2 = math.sqrt((1.0 + x))
	tmp = 0
	if z <= 0.88:
		tmp = (t_2 + (math.sqrt((1.0 + z)) + t_1)) - math.sqrt(x)
	else:
		tmp = (t_2 + t_1) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))
	t_2 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (z <= 0.88)
		tmp = Float64(Float64(t_2 + Float64(sqrt(Float64(1.0 + z)) + t_1)) - sqrt(x));
	else
		tmp = Float64(Float64(t_2 + t_1) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / (sqrt(y) + sqrt((1.0 + y)));
	t_2 = sqrt((1.0 + x));
	tmp = 0.0;
	if (z <= 0.88)
		tmp = (t_2 + (sqrt((1.0 + z)) + t_1)) - sqrt(x);
	else
		tmp = (t_2 + t_1) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 0.88], N[(N[(t$95$2 + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < 0.880000000000000004

   1. Initial program 98.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 98.7%

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in x around inf

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

       1. lift-sqrt.f64 72.3

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

   11. Applied rewrites 72.3%

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

   if 0.880000000000000004 < z

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-- N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 88.8%

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

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 12.0%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-sqrt.f64 88.0

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

   11. Applied rewrites 88.0%

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

Alternative 11: 66.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. flip-- N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 91.9%

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

5. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

6. Taylor expanded in t around inf

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in z around inf

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

    1. lower--.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lift-sqrt.f64 N/A

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

    4. lift-+.f64 N/A

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

    5. lift-sqrt.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-+.f64 N/A

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

    8. lift-+.f64 N/A

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

    9. lift-/.f64 N/A

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

    10. lift-sqrt.f64 66.9

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

11. Applied rewrites 66.9%

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

Alternative 12: 36.0% accurate, 2.1× speedup

Math

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

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 (+ 1.0 x)) (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((1.0 + x)) - 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((1.0d0 + x)) - 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((1.0 + x)) - 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((1.0 + x)) - 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(1.0 + x)) - 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((1.0 + x)) - 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[(1.0 + x), $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 91.8%

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

2. Taylor expanded in z around 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. associate--r+ N/A

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

   2. lower--.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 47.5

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

4. Applied rewrites 47.5%

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

5. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 36.0

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

7. Applied rewrites 36.0%

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

Alternative 13: 9.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. flip-- N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 91.9%

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

5. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

6. Taylor expanded in t around inf

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in z around inf

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

    1. lift-sqrt.f64 9.2

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

11. Applied rewrites 9.2%

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

Alternative 14: 6.2% accurate, 2.5× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt y) (sqrt y)) (* 0.5 (/ 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) - sqrt(y)) + (0.5 * (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) - sqrt(y)) + (0.5d0 * (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) - Math.sqrt(y)) + (0.5 * (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) - math.sqrt(y)) + (0.5 * (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(y) - sqrt(y)) + Float64(0.5 * Float64(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) - sqrt(y)) + (0.5 * (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[y], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.8%

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

2. Taylor expanded in z around 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. associate--r+ N/A

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

   2. lower--.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 47.5

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

4. Applied rewrites 47.5%

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

5. Taylor expanded in y around inf

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

   1. lift-sqrt.f64 4.2

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

7. Applied rewrites 4.2%

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

8. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. sqrt-div N/A

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

   3. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 6.2

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

10. Applied rewrites 6.2%

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

Alternative 15: 3.1% accurate, 3.8× speedup

Math

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

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 z) (+ (sqrt x) (sqrt z))))

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. flip-- N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 91.9%

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

5. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

6. Taylor expanded in t around inf

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in z around inf

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

    1. lift-sqrt.f64 3.1

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

11. Applied rewrites 3.1%

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

Alternative 16: 1.5% accurate, 3.8× speedup

Math

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

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) (+ (sqrt x) (sqrt z))))

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. flip-- N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 91.9%

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

5. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\frac{{\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)}^{3} + {\left(\sqrt{t + 1} - \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right), \left(\sqrt{t + 1} - \sqrt{t}\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right) - \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) \cdot \left(\sqrt{t + 1} - \sqrt{t}\right)\right)}} \]

6. Taylor expanded in t around inf

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in x around inf

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

    1. lift-sqrt.f64 1.5

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

11. Applied rewrites 1.5%

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

Reproduce

herbie shell --seed 2025120 
(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))))