Main:z from

Percentage Accurate: 91.8% → 98.1%

Time: 17.4s

Alternatives: 19

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\\ t_2 := \sqrt{t + 1}\\ t_3 := t\_2 - \sqrt{t}\\ t_4 := t\_1 + t\_3\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_4 \leq 1.999999999999996:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{t\_2 + \sqrt{t}}\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
          (- (sqrt (+ z 1.0)) (sqrt z))))
        (t_2 (sqrt (+ t 1.0)))
        (t_3 (- t_2 (sqrt t)))
        (t_4 (+ t_1 t_3)))
   (if (<= t_4 2e-5)
     (+
      (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
      (- (sqrt t) (sqrt t)))
     (if (<= t_4 1.999999999999996)
       (+
        (- (+ (sqrt (+ 1.0 x)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x))
        t_3)
       (+ t_1 (/ 1.0 (+ t_2 (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z));
	double t_2 = sqrt((t + 1.0));
	double t_3 = t_2 - sqrt(t);
	double t_4 = t_1 + t_3;
	double tmp;
	if (t_4 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	} else if (t_4 <= 1.999999999999996) {
		tmp = ((sqrt((1.0 + x)) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_3;
	} else {
		tmp = t_1 + (1.0 / (t_2 + sqrt(t)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z)))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = Float64(t_2 - sqrt(t))
	t_4 = Float64(t_1 + t_3)
	tmp = 0.0
	if (t_4 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_4 <= 1.999999999999996)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_3);
	else
		tmp = Float64(t_1 + Float64(1.0 / Float64(t_2 + 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[(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]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.999999999999996], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(t$95$1 + N[(1.0 / N[(t$95$2 + 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))) < 2.00000000000000016e-5

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-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))) < 1.999999999999996

   1. Initial program 94.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 95.4%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 98.1

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

   6. Applied rewrites 98.1%

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

   if 1.999999999999996 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      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 97.8%

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

   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 98.4%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_1\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_2 \leq 1.999999999999996:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_2
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          t_1)))
   (if (<= t_2 2e-5)
     (+
      (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
      (- (sqrt t) (sqrt t)))
     (if (<= t_2 1.999999999999996)
       (+
        (- (+ (sqrt (+ 1.0 x)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x))
        t_1)
       t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_1;
	double tmp;
	if (t_2 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	} else if (t_2 <= 1.999999999999996) {
		tmp = ((sqrt((1.0 + x)) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_1)
	tmp = 0.0
	if (t_2 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_2 <= 1.999999999999996)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.999999999999996], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$2]]]]

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))) < 2.00000000000000016e-5

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-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))) < 1.999999999999996

   1. Initial program 94.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 95.4%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 98.1

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

   6. Applied rewrites 98.1%

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

   if 1.999999999999996 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.7%

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

Alternative 3: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t} - \sqrt{t}\\ t_2 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ t_3 := t\_2 + \left(\sqrt{z + 1} - \sqrt{z}\right)\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := t\_3 + t\_4\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + t\_1\\ \mathbf{elif}\;t\_5 \leq 1.999999999999996:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.99999995:\\ \;\;\;\;t\_3 + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \left(1 - \sqrt{z}\right)\right) + t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt t) (sqrt t)))
        (t_2 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))))
        (t_3 (+ t_2 (- (sqrt (+ z 1.0)) (sqrt z))))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5 (+ t_3 t_4)))
   (if (<= t_5 2e-5)
     (+ (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y)))) t_1)
     (if (<= t_5 1.999999999999996)
       (+
        (- (+ (sqrt (+ 1.0 x)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x))
        t_4)
       (if (<= t_5 2.99999995)
         (+ t_3 t_1)
         (+ (+ t_2 (- 1.0 (sqrt z))) t_4))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(t) - sqrt(t);
	double t_2 = (sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y));
	double t_3 = t_2 + (sqrt((z + 1.0)) - sqrt(z));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = t_3 + t_4;
	double tmp;
	if (t_5 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + t_1;
	} else if (t_5 <= 1.999999999999996) {
		tmp = ((sqrt((1.0 + x)) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_4;
	} else if (t_5 <= 2.99999995) {
		tmp = t_3 + t_1;
	} else {
		tmp = (t_2 + (1.0 - sqrt(z))) + t_4;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(t) - sqrt(t))
	t_2 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y)))
	t_3 = Float64(t_2 + Float64(sqrt(Float64(z + 1.0)) - sqrt(z)))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(t_3 + t_4)
	tmp = 0.0
	if (t_5 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + t_1);
	elseif (t_5 <= 1.999999999999996)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_4);
	elseif (t_5 <= 2.99999995)
		tmp = Float64(t_3 + t_1);
	else
		tmp = Float64(Float64(t_2 + Float64(1.0 - sqrt(z))) + t_4);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$5, 1.999999999999996], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.99999995], N[(t$95$3 + t$95$1), $MachinePrecision], N[(N[(t$95$2 + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-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))) < 1.999999999999996

   1. Initial program 94.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 95.4%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 98.1

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

   6. Applied rewrites 98.1%

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

   if 1.999999999999996 < (+.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))) < 2.99999994999999986

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 96.7%

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

   if 2.99999994999999986 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 98.5%

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

Alternative 4: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \left(t\_3 + \left(t\_2 - \sqrt{z}\right)\right) + t\_4\\ t_6 := \sqrt{1 + x}\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\left(\left(t\_6 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 2.99999995:\\ \;\;\;\;\left(\left(\left(t\_6 + t\_1\right) + t\_2\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 + \left(1 - \sqrt{z}\right)\right) + t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y))))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5 (+ (+ t_3 (- t_2 (sqrt z))) t_4))
        (t_6 (sqrt (+ 1.0 x))))
   (if (<= t_5 2e-5)
     (+
      (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
      (- (sqrt t) (sqrt t)))
     (if (<= t_5 2.0)
       (+ (- (+ t_6 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x)) t_4)
       (if (<= t_5 2.99999995)
         (- (- (+ (+ t_6 t_1) t_2) (sqrt x)) (+ (sqrt z) (sqrt y)))
         (+ (+ t_3 (- 1.0 (sqrt z))) t_4))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = (sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = (t_3 + (t_2 - sqrt(z))) + t_4;
	double t_6 = sqrt((1.0 + x));
	double tmp;
	if (t_5 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	} else if (t_5 <= 2.0) {
		tmp = ((t_6 + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_4;
	} else if (t_5 <= 2.99999995) {
		tmp = (((t_6 + t_1) + t_2) - sqrt(x)) - (sqrt(z) + sqrt(y));
	} else {
		tmp = (t_3 + (1.0 - sqrt(z))) + t_4;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y)))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(Float64(t_3 + Float64(t_2 - sqrt(z))) + t_4)
	t_6 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_5 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(Float64(t_6 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_4);
	elseif (t_5 <= 2.99999995)
		tmp = Float64(Float64(Float64(Float64(t_6 + t_1) + t_2) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(Float64(t_3 + Float64(1.0 - sqrt(z))) + t_4);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(N[(t$95$6 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 2.99999995], N[(N[(N[(N[(t$95$6 + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-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

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. 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 96.7%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 98.1

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

   6. Applied rewrites 98.1%

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

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.99999994999999986

   1. Initial program 91.5%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 90.6%

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

   if 2.99999994999999986 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 98.5%

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

Alternative 5: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_2 - \sqrt{z}\right)\right) + t\_4\\ t_6 := \sqrt{1 + x}\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\left(\left(t\_6 + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 3:\\ \;\;\;\;\left(\left(\left(t\_6 + t\_1\right) + t\_2\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_6 + t\_2\right) + t\_3\right) + 1\right) - \sqrt{t}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (- t_3 (sqrt t)))
        (t_5
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_2 (sqrt z)))
          t_4))
        (t_6 (sqrt (+ 1.0 x))))
   (if (<= t_5 2e-5)
     (+
      (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
      (- (sqrt t) (sqrt t)))
     (if (<= t_5 2.0)
       (+ (- (+ t_6 (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x)) t_4)
       (if (<= t_5 3.0)
         (- (- (+ (+ t_6 t_1) t_2) (sqrt x)) (+ (sqrt z) (sqrt y)))
         (- (- (+ (+ (+ t_6 t_2) t_3) 1.0) (sqrt t)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 - sqrt(t);
	double t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_2 - sqrt(z))) + t_4;
	double t_6 = sqrt((1.0 + x));
	double tmp;
	if (t_5 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	} else if (t_5 <= 2.0) {
		tmp = ((t_6 + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_4;
	} else if (t_5 <= 3.0) {
		tmp = (((t_6 + t_1) + t_2) - sqrt(x)) - (sqrt(z) + sqrt(y));
	} else {
		tmp = ((((t_6 + t_2) + t_3) + 1.0) - sqrt(t)) - sqrt(x);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_2 - sqrt(z))) + t_4)
	t_6 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_5 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(Float64(t_6 + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_4);
	elseif (t_5 <= 3.0)
		tmp = Float64(Float64(Float64(Float64(t_6 + t_1) + t_2) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_6 + t_2) + t_3) + 1.0) - sqrt(t)) - 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(N[(t$95$6 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 3.0], N[(N[(N[(N[(t$95$6 + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$6 + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-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

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. 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 96.7%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 98.1

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

   6. Applied rewrites 98.1%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 97.2%

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

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

   1. Initial program 97.2%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around inf

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

      1. lift-sqrt.f64 95.7

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

   7. Applied rewrites 95.7%

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

Alternative 6: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{x + 1} - \sqrt{x}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(t\_2 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_1\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_4 \leq 1.999999999998:\\ \;\;\;\;\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 + \frac{1}{1 + \sqrt{y}}\right) + t\_1\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (+ (+ (+ t_2 (- (sqrt (+ y 1.0)) (sqrt y))) t_1) t_3)))
   (if (<= t_4 2e-5)
     (+
      (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
      (- (sqrt t) (sqrt t)))
     (if (<= t_4 1.999999999998)
       (+
        (- (+ (sqrt (+ 1.0 x)) (/ 1.0 (+ (sqrt y) (sqrt (+ 1.0 y))))) (sqrt x))
        t_3)
       (+ (+ (+ t_2 (/ 1.0 (+ 1.0 (sqrt y)))) t_1) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((x + 1.0)) - sqrt(x);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = ((t_2 + (sqrt((y + 1.0)) - sqrt(y))) + t_1) + t_3;
	double tmp;
	if (t_4 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	} else if (t_4 <= 1.999999999998) {
		tmp = ((sqrt((1.0 + x)) + (1.0 / (sqrt(y) + sqrt((1.0 + y))))) - sqrt(x)) + t_3;
	} else {
		tmp = ((t_2 + (1.0 / (1.0 + sqrt(y)))) + t_1) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(t_2 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_1) + t_3)
	tmp = 0.0
	if (t_4 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_4 <= 1.999999999998)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) + Float64(1.0 / Float64(sqrt(y) + sqrt(Float64(1.0 + y))))) - sqrt(x)) + t_3);
	else
		tmp = Float64(Float64(Float64(t_2 + Float64(1.0 / Float64(1.0 + sqrt(y)))) + t_1) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.999999999998], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(t$95$2 + N[(1.0 / N[(1.0 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $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))) < 2.00000000000000016e-5

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-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))) < 1.99999999999800004

   1. Initial program 94.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 95.2%

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

   4. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-sqrt.f64 98.0

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

   6. Applied rewrites 98.0%

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

   if 1.99999999999800004 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 97.7%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 97.6

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

   6. Applied rewrites 97.6%

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

Alternative 7: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := 0.5 \cdot \frac{1}{\sqrt{y}}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{t + 1}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_5\\ t_7 := \sqrt{1 + x}\\ \mathbf{if}\;t\_6 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_6 \leq 1.0002:\\ \;\;\;\;\left(\left(t\_7 + t\_2\right) - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\left(\left(1 + t\_1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(\left(\left(t\_7 + t\_1\right) + t\_3\right) - \sqrt{x}\right) - \left(\sqrt{z} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_7 + t\_3\right) + t\_4\right) + 1\right) - \sqrt{t}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (* 0.5 (/ 1.0 (sqrt y))))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (- t_4 (sqrt t)))
        (t_6
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          t_5))
        (t_7 (sqrt (+ 1.0 x))))
   (if (<= t_6 2e-5)
     (+ (fma 0.5 (/ 1.0 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_6 1.0002)
       (+ (- (+ t_7 t_2) (sqrt x)) (* 0.5 (/ 1.0 (sqrt t))))
       (if (<= t_6 2.0)
         (+ (- (- (+ 1.0 t_1) (sqrt x)) (sqrt y)) t_5)
         (if (<= t_6 3.0)
           (- (- (+ (+ t_7 t_1) t_3) (sqrt x)) (+ (sqrt z) (sqrt y)))
           (- (- (+ (+ (+ t_7 t_3) t_4) 1.0) (sqrt t)) (sqrt x))))))))

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 = 0.5 * (1.0 / sqrt(y));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0));
	double t_5 = t_4 - sqrt(t);
	double t_6 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + t_5;
	double t_7 = sqrt((1.0 + x));
	double tmp;
	if (t_6 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), t_2) + (sqrt(t) - sqrt(t));
	} else if (t_6 <= 1.0002) {
		tmp = ((t_7 + t_2) - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	} else if (t_6 <= 2.0) {
		tmp = (((1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_5;
	} else if (t_6 <= 3.0) {
		tmp = (((t_7 + t_1) + t_3) - sqrt(x)) - (sqrt(z) + sqrt(y));
	} else {
		tmp = ((((t_7 + t_3) + t_4) + 1.0) - sqrt(t)) - sqrt(x);
	}
	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(0.5 * Float64(1.0 / sqrt(y)))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + t_5)
	t_7 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_6 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_6 <= 1.0002)
		tmp = Float64(Float64(Float64(t_7 + t_2) - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_5);
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(Float64(Float64(t_7 + t_1) + t_3) - sqrt(x)) - Float64(sqrt(z) + sqrt(y)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_7 + t_3) + t_4) + 1.0) - sqrt(t)) - 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.0002], N[(N[(N[(t$95$7 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(N[(N[(t$95$7 + 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[(N[(N[(N[(t$95$7 + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 94.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 20.4

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

   4. Applied rewrites 20.4%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 98.6

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

   10. Applied rewrites 98.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in 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 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.2%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in t around inf

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 97.2%

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

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

   1. Initial program 97.2%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around inf

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

      1. lift-sqrt.f64 95.7

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

   7. Applied rewrites 95.7%

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

Alternative 8: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := 0.5 \cdot \frac{1}{\sqrt{y}}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{t + 1}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_5\\ t_7 := \sqrt{1 + x}\\ \mathbf{if}\;t\_6 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_6 \leq 1.0002:\\ \;\;\;\;\left(\left(t\_7 + t\_2\right) - \sqrt{x}\right) + 0.5 \cdot \frac{1}{\sqrt{t}}\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\left(\left(1 + t\_1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(1 + \left(t\_7 + \sqrt{1 + z}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_7 + t\_3\right) + t\_4\right) + 1\right) - \sqrt{t}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (* 0.5 (/ 1.0 (sqrt y))))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (- t_4 (sqrt t)))
        (t_6
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          t_5))
        (t_7 (sqrt (+ 1.0 x))))
   (if (<= t_6 2e-5)
     (+ (fma 0.5 (/ 1.0 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_6 1.0002)
       (+ (- (+ t_7 t_2) (sqrt x)) (* 0.5 (/ 1.0 (sqrt t))))
       (if (<= t_6 2.0)
         (+ (- (- (+ 1.0 t_1) (sqrt x)) (sqrt y)) t_5)
         (if (<= t_6 3.0)
           (-
            (+ 1.0 (+ t_7 (sqrt (+ 1.0 z))))
            (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
           (- (- (+ (+ (+ t_7 t_3) t_4) 1.0) (sqrt t)) (sqrt x))))))))

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 = 0.5 * (1.0 / sqrt(y));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0));
	double t_5 = t_4 - sqrt(t);
	double t_6 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + t_5;
	double t_7 = sqrt((1.0 + x));
	double tmp;
	if (t_6 <= 2e-5) {
		tmp = fma(0.5, (1.0 / sqrt(x)), t_2) + (sqrt(t) - sqrt(t));
	} else if (t_6 <= 1.0002) {
		tmp = ((t_7 + t_2) - sqrt(x)) + (0.5 * (1.0 / sqrt(t)));
	} else if (t_6 <= 2.0) {
		tmp = (((1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_5;
	} else if (t_6 <= 3.0) {
		tmp = (1.0 + (t_7 + sqrt((1.0 + z)))) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	} else {
		tmp = ((((t_7 + t_3) + t_4) + 1.0) - sqrt(t)) - sqrt(x);
	}
	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(0.5 * Float64(1.0 / sqrt(y)))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + t_5)
	t_7 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_6 <= 2e-5)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_6 <= 1.0002)
		tmp = Float64(Float64(Float64(t_7 + t_2) - sqrt(x)) + Float64(0.5 * Float64(1.0 / sqrt(t))));
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_5);
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(1.0 + Float64(t_7 + sqrt(Float64(1.0 + z)))) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_7 + t_3) + t_4) + 1.0) - sqrt(t)) - 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 2e-5], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.0002], N[(N[(N[(t$95$7 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(1.0 + N[(t$95$7 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$7 + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

   if 2.00000000000000016e-5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 94.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 20.4

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

   4. Applied rewrites 20.4%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 98.6

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

   10. Applied rewrites 98.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in 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 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.2%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 5.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 96.9

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

   7. Applied rewrites 96.9%

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

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

   1. Initial program 97.2%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around inf

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

      1. lift-sqrt.f64 95.7

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

   7. Applied rewrites 95.7%

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

Alternative 9: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\\ \mathbf{if}\;x \leq 880000000:\\ \;\;\;\;\frac{{t\_2}^{3} + {t\_1}^{3}}{\mathsf{fma}\left(t\_2, t\_2, t\_1 \cdot t\_1 - t\_2 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_2
         (+
          (+
           (- (sqrt (+ x 1.0)) (sqrt x))
           (/ (- (+ y 1.0) y) (+ (sqrt (+ y 1.0)) (sqrt y))))
          (- (sqrt (+ z 1.0)) (sqrt z)))))
   (if (<= x 880000000.0)
     (/
      (+ (pow t_2 3.0) (pow t_1 3.0))
      (fma t_2 t_2 (- (* t_1 t_1) (* t_2 t_1))))
     (+
      (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
      (- (sqrt t) (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = ((sqrt((x + 1.0)) - sqrt(x)) + (((y + 1.0) - y) / (sqrt((y + 1.0)) + sqrt(y)))) + (sqrt((z + 1.0)) - sqrt(z));
	double tmp;
	if (x <= 880000000.0) {
		tmp = (pow(t_2, 3.0) + pow(t_1, 3.0)) / fma(t_2, t_2, ((t_1 * t_1) - (t_2 * t_1)));
	} else {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(Float64(Float64(y + 1.0) - y) / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z)))
	tmp = 0.0
	if (x <= 880000000.0)
		tmp = Float64(Float64((t_2 ^ 3.0) + (t_1 ^ 3.0)) / fma(t_2, t_2, Float64(Float64(t_1 * t_1) - Float64(t_2 * t_1))));
	else
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + 1.0), $MachinePrecision] - y), $MachinePrecision] / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 880000000.0], N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8e8

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. 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 96.9%

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

   5. Applied rewrites 97.3%

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

   if 8.8e8 < x

   1. Initial program 7.4%

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

   2. Taylor expanded in z around inf

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

      1. 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 3.6

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

   4. Applied rewrites 3.6%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.9%

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

Alternative 10: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := 0.5 \cdot \frac{1}{\sqrt{y}}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{t + 1}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_5\\ t_7 := \sqrt{1 + x}\\ \mathbf{if}\;t\_6 \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_6 \leq 1.0002:\\ \;\;\;\;\left(\left(1 + t\_2\right) - \sqrt{x}\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\left(\left(1 + t\_1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(1 + \left(t\_7 + \sqrt{1 + z}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_7 + t\_3\right) + t\_4\right) + 1\right) - \sqrt{t}\right) - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (* 0.5 (/ 1.0 (sqrt y))))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (- t_4 (sqrt t)))
        (t_6
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- t_3 (sqrt z)))
          t_5))
        (t_7 (sqrt (+ 1.0 x))))
   (if (<= t_6 0.4)
     (+ (fma 0.5 (/ 1.0 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_6 1.0002)
       (+ (- (+ 1.0 t_2) (sqrt x)) t_5)
       (if (<= t_6 2.0)
         (+ (- (- (+ 1.0 t_1) (sqrt x)) (sqrt y)) t_5)
         (if (<= t_6 3.0)
           (-
            (+ 1.0 (+ t_7 (sqrt (+ 1.0 z))))
            (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
           (- (- (+ (+ (+ t_7 t_3) t_4) 1.0) (sqrt t)) (sqrt x))))))))

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 = 0.5 * (1.0 / sqrt(y));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((t + 1.0));
	double t_5 = t_4 - sqrt(t);
	double t_6 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (t_3 - sqrt(z))) + t_5;
	double t_7 = sqrt((1.0 + x));
	double tmp;
	if (t_6 <= 0.4) {
		tmp = fma(0.5, (1.0 / sqrt(x)), t_2) + (sqrt(t) - sqrt(t));
	} else if (t_6 <= 1.0002) {
		tmp = ((1.0 + t_2) - sqrt(x)) + t_5;
	} else if (t_6 <= 2.0) {
		tmp = (((1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_5;
	} else if (t_6 <= 3.0) {
		tmp = (1.0 + (t_7 + sqrt((1.0 + z)))) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	} else {
		tmp = ((((t_7 + t_3) + t_4) + 1.0) - sqrt(t)) - sqrt(x);
	}
	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(0.5 * Float64(1.0 / sqrt(y)))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_1 - sqrt(y))) + Float64(t_3 - sqrt(z))) + t_5)
	t_7 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (t_6 <= 0.4)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_6 <= 1.0002)
		tmp = Float64(Float64(Float64(1.0 + t_2) - sqrt(x)) + t_5);
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_5);
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(1.0 + Float64(t_7 + sqrt(Float64(1.0 + z)))) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_7 + t_3) + t_4) + 1.0) - sqrt(t)) - 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, 0.4], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.0002], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(1.0 + N[(t$95$7 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$7 + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 15.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 5.3

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 16.0

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 89.4%

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

   if 0.40000000000000002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf

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

      1. 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 20.5

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

   4. Applied rewrites 20.5%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in 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 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.2%

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

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

   1. Initial program 97.4%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 5.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 96.9

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

   7. Applied rewrites 96.9%

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

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

   1. Initial program 97.2%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in x around inf

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

      1. lift-sqrt.f64 95.7

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

   7. Applied rewrites 95.7%

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

Alternative 11: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := 0.5 \cdot \frac{1}{\sqrt{y}}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_4 \leq 1.0002:\\ \;\;\;\;\left(\left(1 + t\_2\right) - \sqrt{x}\right) + t\_3\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\left(\left(\left(1 + t\_1\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (* 0.5 (/ 1.0 (sqrt y))))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_1 (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          t_3)))
   (if (<= t_4 0.4)
     (+ (fma 0.5 (/ 1.0 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_4 1.0002)
       (+ (- (+ 1.0 t_2) (sqrt x)) t_3)
       (if (<= t_4 2.0)
         (+ (- (- (+ 1.0 t_1) (sqrt x)) (sqrt y)) t_3)
         (-
          (+ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt (+ 1.0 z))))
          (+ (+ (sqrt z) (sqrt y)) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = 0.5 * (1.0 / sqrt(y));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = (((sqrt((x + 1.0)) - sqrt(x)) + (t_1 - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + t_3;
	double tmp;
	if (t_4 <= 0.4) {
		tmp = fma(0.5, (1.0 / sqrt(x)), t_2) + (sqrt(t) - sqrt(t));
	} else if (t_4 <= 1.0002) {
		tmp = ((1.0 + t_2) - sqrt(x)) + t_3;
	} else if (t_4 <= 2.0) {
		tmp = (((1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_3;
	} else {
		tmp = (1.0 + (sqrt((1.0 + x)) + sqrt((1.0 + z)))) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	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(0.5 * Float64(1.0 / sqrt(y)))
	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))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + t_3)
	tmp = 0.0
	if (t_4 <= 0.4)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_4 <= 1.0002)
		tmp = Float64(Float64(Float64(1.0 + t_2) - sqrt(x)) + t_3);
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_1) - sqrt(x)) - sqrt(y)) + t_3);
	else
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + x)) + sqrt(Float64(1.0 + z)))) - Float64(Float64(sqrt(z) + sqrt(y)) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 0.4], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.0002], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(N[(N[(1.0 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 15.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 5.3

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 16.0

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 89.4%

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

   if 0.40000000000000002 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.0002

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf

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

      1. 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 20.5

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

   4. Applied rewrites 20.5%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.6%

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

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

   1. Initial program 97.7%

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

   2. Taylor expanded in 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 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.2%

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

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

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 80.9

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

   7. Applied rewrites 80.9%

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

Alternative 12: 70.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := 0.5 \cdot \frac{1}{\sqrt{y}}\\ t_3 := \sqrt{y + 1}\\ t_4 := \left(\sqrt{x + 1} - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\\ \mathbf{if}\;t\_4 \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, t\_2\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_4 \leq 1.0002:\\ \;\;\;\;\left(\left(1 + t\_2\right) - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 + t\_3\right) - \sqrt{x}\right) - \sqrt{y}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_2 (* 0.5 (/ 1.0 (sqrt y))))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- t_3 (sqrt y)))))
   (if (<= t_4 0.4)
     (+ (fma 0.5 (/ 1.0 (sqrt x)) t_2) (- (sqrt t) (sqrt t)))
     (if (<= t_4 1.0002)
       (+ (- (+ 1.0 t_2) (sqrt x)) t_1)
       (+ (- (- (+ 1.0 t_3) (sqrt x)) (sqrt y)) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = 0.5 * (1.0 / sqrt(y));
	double t_3 = sqrt((y + 1.0));
	double t_4 = (sqrt((x + 1.0)) - sqrt(x)) + (t_3 - sqrt(y));
	double tmp;
	if (t_4 <= 0.4) {
		tmp = fma(0.5, (1.0 / sqrt(x)), t_2) + (sqrt(t) - sqrt(t));
	} else if (t_4 <= 1.0002) {
		tmp = ((1.0 + t_2) - sqrt(x)) + t_1;
	} else {
		tmp = (((1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = Float64(0.5 * Float64(1.0 / sqrt(y)))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(t_3 - sqrt(y)))
	tmp = 0.0
	if (t_4 <= 0.4)
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), t_2) + Float64(sqrt(t) - sqrt(t)));
	elseif (t_4 <= 1.0002)
		tmp = Float64(Float64(Float64(1.0 + t_2) - sqrt(x)) + t_1);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + t_3) - sqrt(x)) - sqrt(y)) + t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 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$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.4], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.0002], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(1.0 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 0.40000000000000002

   1. Initial program 15.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 5.3

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 16.0

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 89.4

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

   10. Applied rewrites 89.4%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 89.4%

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

   if 0.40000000000000002 < (+.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))) < 1.0002

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf

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

      1. 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 20.5

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

   4. Applied rewrites 20.5%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.6%

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

   if 1.0002 < (+.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)))

   1. Initial program 97.5%

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

   2. Taylor expanded in 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 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

Alternative 13: 40.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 410000000:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 410000000.0)
   (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- (sqrt (+ t 1.0)) (sqrt t)))
   (+
    (fma 0.5 (/ 1.0 (sqrt x)) (* 0.5 (/ 1.0 (sqrt y))))
    (- (sqrt t) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 410000000.0) {
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = fma(0.5, (1.0 / sqrt(x)), (0.5 * (1.0 / sqrt(y)))) + (sqrt(t) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 410000000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(fma(0.5, Float64(1.0 / sqrt(x)), Float64(0.5 * Float64(1.0 / sqrt(y)))) + Float64(sqrt(t) - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 410000000.0], 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], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1e8

   1. Initial program 96.7%

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

   2. 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 50.1

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

   4. Applied rewrites 50.1%

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

   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 37.5

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

   7. Applied rewrites 37.5%

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

   if 4.1e8 < x

   1. Initial program 7.6%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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 3.7

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

   4. Applied rewrites 3.7%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.7

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.7

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

   10. Applied rewrites 95.7%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 95.7%

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

Alternative 14: 39.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 68000000:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\sqrt{x}} + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= x 68000000.0)
     (+ (- (sqrt (+ 1.0 x)) (sqrt x)) t_1)
     (+ (* 0.5 (/ 1.0 (sqrt x))) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (x <= 68000000.0) {
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + t_1;
	} else {
		tmp = (0.5 * (1.0 / sqrt(x))) + t_1;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    if (x <= 68000000.0d0) then
        tmp = (sqrt((1.0d0 + x)) - sqrt(x)) + t_1
    else
        tmp = (0.5d0 * (1.0d0 / sqrt(x))) + t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (x <= 68000000.0) {
		tmp = (Math.sqrt((1.0 + x)) - Math.sqrt(x)) + t_1;
	} else {
		tmp = (0.5 * (1.0 / Math.sqrt(x))) + t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if x <= 68000000.0:
		tmp = (math.sqrt((1.0 + x)) - math.sqrt(x)) + t_1
	else:
		tmp = (0.5 * (1.0 / math.sqrt(x))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (x <= 68000000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + t_1);
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (x <= 68000000.0)
		tmp = (sqrt((1.0 + x)) - sqrt(x)) + t_1;
	else
		tmp = (0.5 * (1.0 / sqrt(x))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 68000000.0], N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.8e7

   1. Initial program 96.7%

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

   2. 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 50.1

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

   4. Applied rewrites 50.1%

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

   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 37.5

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

   7. Applied rewrites 37.5%

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

   if 6.8e7 < x

   1. Initial program 7.8%

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

   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 3.8

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 7.9

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

   7. Applied rewrites 7.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 95.6

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

   10. Applied rewrites 95.6%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-/.f64 81.6

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

   13. Applied rewrites 81.6%

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

Alternative 15: 15.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+32}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\sqrt{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.25e+32)
   (- (+ (sqrt (+ 1.0 y)) (sqrt z)) (+ (sqrt y) (sqrt z)))
   (+ (* 0.5 (/ 1.0 (sqrt x))) (- (sqrt (+ t 1.0)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.25e+32) {
		tmp = (sqrt((1.0 + y)) + sqrt(z)) - (sqrt(y) + sqrt(z));
	} else {
		tmp = (0.5 * (1.0 / sqrt(x))) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.25d+32) then
        tmp = (sqrt((1.0d0 + y)) + sqrt(z)) - (sqrt(y) + sqrt(z))
    else
        tmp = (0.5d0 * (1.0d0 / sqrt(x))) + (sqrt((t + 1.0d0)) - sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.25e+32) {
		tmp = (Math.sqrt((1.0 + y)) + Math.sqrt(z)) - (Math.sqrt(y) + Math.sqrt(z));
	} else {
		tmp = (0.5 * (1.0 / Math.sqrt(x))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 1.25e+32:
		tmp = (math.sqrt((1.0 + y)) + math.sqrt(z)) - (math.sqrt(y) + math.sqrt(z))
	else:
		tmp = (0.5 * (1.0 / math.sqrt(x))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.25e+32)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + sqrt(z)) - Float64(sqrt(y) + sqrt(z)));
	else
		tmp = Float64(Float64(0.5 * Float64(1.0 / sqrt(x))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.25e+32)
		tmp = (sqrt((1.0 + y)) + sqrt(z)) - (sqrt(y) + sqrt(z));
	else
		tmp = (0.5 * (1.0 / sqrt(x))) + (sqrt((t + 1.0)) - sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, 1.25e+32], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2499999999999999e32

   1. Initial program 95.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. 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 95.6%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   6. Applied rewrites 19.8%

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

   7. Taylor expanded in z around inf

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

      1. lift-sqrt.f64 17.8

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

   9. Applied rewrites 17.8%

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

   if 1.2499999999999999e32 < z

   1. Initial program 89.5%

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

   2. Taylor expanded in 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 59.6

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 44.5

          \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 44.5%

      \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \frac{1}{\sqrt{y}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\sqrt{1}}{\sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{1}{\sqrt{x}}, \frac{1}{2} \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-*.f64 14.8

          \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{\sqrt{x}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 14.8%

       \[\leadsto \mathsf{fma}\left(0.5, \frac{1}{\color{blue}{\sqrt{x}}}, 0.5 \cdot \frac{1}{\sqrt{y}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. sqrt-div N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\sqrt{1}}{\sqrt{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \frac{1}{\sqrt{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{1}{\sqrt{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       5. lift-/.f64 13.5

          \[\leadsto 0.5 \cdot \frac{1}{\sqrt{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. Applied rewrites 13.5%

       \[\leadsto 0.5 \cdot \frac{1}{\sqrt{x}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 8.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + y} + \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{z}\right) \end{array} \]

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 y)) (sqrt z)) (+ (sqrt y) (sqrt z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((1.0 + y)) + sqrt(z)) - (sqrt(y) + 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 + y)) + sqrt(z)) - (sqrt(y) + 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 + y)) + Math.sqrt(z)) - (Math.sqrt(y) + Math.sqrt(z));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + y)) + math.sqrt(z)) - (math.sqrt(y) + 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 + y)) + sqrt(z)) - Float64(sqrt(y) + 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 + y)) + sqrt(z)) - (sqrt(y) + 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 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $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

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

   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 91.9%

   \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]
5. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)} \]

6. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)} \]

7. Taylor expanded in z around inf

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
8. Step-by-step derivation

   1. lift-sqrt.f64 8.9

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]

9. Applied rewrites 8.9%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{z}\right) \]
10. Add Preprocessing

Alternative 17: 4.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{y} - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt y) (sqrt y)) (- (sqrt (+ t 1.0)) (sqrt t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt(y) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt(y) - sqrt(y)) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (Math.sqrt(y) - Math.sqrt(y)) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt(y) - math.sqrt(y)) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(y) - sqrt(y)) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt(y) - sqrt(y)) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

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 47.6

       \[\leadsto \left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied rewrites 47.6%

   \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} + \sqrt{y + 1}\right) - \sqrt{x}\right) - \sqrt{y}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

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. Add Preprocessing

Alternative 18: 3.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{t} - \sqrt{t} \end{array} \]

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 t) (sqrt t)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(t) - sqrt(t);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(t) - sqrt(t)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(t) - Math.sqrt(t);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(t) - math.sqrt(t)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(t) - sqrt(t))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(t) - sqrt(t);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

3. Applied rewrites 11.4%

   \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + x} - \left(\sqrt{x} - \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \sqrt{t + 1}\right) - \sqrt{t}} \]

4. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\sqrt{t}} - \sqrt{t} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 3.1

      \[\leadsto \sqrt{t} - \sqrt{t} \]

6. Applied rewrites 3.1%

   \[\leadsto \color{blue}{\sqrt{t}} - \sqrt{t} \]
7. Add Preprocessing

Alternative 19: 1.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{y} \cdot -1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* (sqrt y) -1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(y) * -1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(y) * (-1.0d0)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(y) * -1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(y) * -1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(sqrt(y) * -1.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(y) * -1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Sqrt[y], $MachinePrecision] * -1.0), $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 y around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate--r+ N/A

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) - \sqrt{t}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right)\right) - \sqrt{t}\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

4. Applied rewrites 10.3%

   \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{1 + x} + \sqrt{z + 1}\right) + \sqrt{t + 1}\right) + 1\right) - \sqrt{t}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

5. Taylor expanded in y around -inf

   \[\leadsto \sqrt{y} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
6. Step-by-step derivation

   1. sqrt-pow2 N/A

      \[\leadsto \sqrt{y} \cdot {-1}^{\left(\frac{2}{\color{blue}{2}}\right)} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{y} \cdot {-1}^{1} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{y} \cdot -1 \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{y} \cdot -1 \]

   5. lift-sqrt.f64 1.7

      \[\leadsto \sqrt{y} \cdot -1 \]

7. Applied rewrites 1.7%

   \[\leadsto \sqrt{y} \cdot \color{blue}{-1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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))))