Main:z from

Percentage Accurate: 91.6% → 97.9%

Time: 15.9s

Alternatives: 19

Speedup: 1.0×

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: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{z}}, 0.5 \cdot \sqrt{z}\right)}{z}\\ t_5 := \sqrt{y + 1} - \sqrt{y}\\ t_6 := \left(\sqrt{x + 1} - \sqrt{x}\right) + t\_5\\ t_7 := \left(t\_6 + t\_2\right) + t\_3\\ \mathbf{if}\;t\_7 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, 0.5 \cdot \sqrt{x}\right)}{x} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_4\right) + t\_1\\ \mathbf{elif}\;t\_7 \leq 2.002:\\ \;\;\;\;\left(t\_6 + t\_4\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_5\right) + t\_2\right) + t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (sqrt (/ 1.0 t))))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (/ (fma -0.125 (sqrt (/ 1.0 z)) (* 0.5 (sqrt z))) z))
        (t_5 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_6 (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_5))
        (t_7 (+ (+ t_6 t_2) t_3)))
   (if (<= t_7 4e-6)
     (+
      (+
       (+
        (/ (fma -0.125 (sqrt (/ 1.0 x)) (* 0.5 (sqrt x))) x)
        (* 0.5 (sqrt (/ 1.0 y))))
       t_4)
      t_1)
     (if (<= t_7 2.002)
       (+ (+ t_6 t_4) t_1)
       (+ (+ (+ (- 1.0 (sqrt x)) t_5) t_2) t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * sqrt((1.0 / t));
	double t_2 = sqrt((z + 1.0)) - sqrt(z);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = fma(-0.125, sqrt((1.0 / z)), (0.5 * sqrt(z))) / z;
	double t_5 = sqrt((y + 1.0)) - sqrt(y);
	double t_6 = (sqrt((x + 1.0)) - sqrt(x)) + t_5;
	double t_7 = (t_6 + t_2) + t_3;
	double tmp;
	if (t_7 <= 4e-6) {
		tmp = (((fma(-0.125, sqrt((1.0 / x)), (0.5 * sqrt(x))) / x) + (0.5 * sqrt((1.0 / y)))) + t_4) + t_1;
	} else if (t_7 <= 2.002) {
		tmp = (t_6 + t_4) + t_1;
	} else {
		tmp = (((1.0 - sqrt(x)) + t_5) + t_2) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(fma(-0.125, sqrt(Float64(1.0 / z)), Float64(0.5 * sqrt(z))) / z)
	t_5 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_6 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_5)
	t_7 = Float64(Float64(t_6 + t_2) + t_3)
	tmp = 0.0
	if (t_7 <= 4e-6)
		tmp = Float64(Float64(Float64(Float64(fma(-0.125, sqrt(Float64(1.0 / x)), Float64(0.5 * sqrt(x))) / x) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + t_4) + t_1);
	elseif (t_7 <= 2.002)
		tmp = Float64(Float64(t_6 + t_4) + t_1);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_5) + t_2) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-0.125 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$6 + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$7, 4e-6], N[(N[(N[(N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$7, 2.002], N[(N[(t$95$6 + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.99999999999999982e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 29.5

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

   10. Applied rewrites 29.5%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lift-sqrt.f64 6.7

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

   13. Applied rewrites 6.7%

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

   if 3.99999999999999982e-6 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.0019999999999998

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if 2.0019999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{z}}, 0.5 \cdot \sqrt{z}\right)}{z}\\ t_4 := \sqrt{y + 1} - \sqrt{y}\\ t_5 := \left(\sqrt{x + 1} - \sqrt{x}\right) + t\_4\\ t_6 := \left(t\_5 + t\_1\right) + t\_2\\ t_7 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;t\_6 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + t\_3\right) + t\_7\\ \mathbf{elif}\;t\_6 \leq 2.002:\\ \;\;\;\;\left(t\_5 + t\_3\right) + t\_7\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_4\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (/ (fma -0.125 (sqrt (/ 1.0 z)) (* 0.5 (sqrt z))) z))
        (t_4 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_5 (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_4))
        (t_6 (+ (+ t_5 t_1) t_2))
        (t_7 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= t_6 4e-6)
     (+ (+ (+ (* 0.5 (sqrt (/ 1.0 x))) (* 0.5 (sqrt (/ 1.0 y)))) t_3) t_7)
     (if (<= t_6 2.002)
       (+ (+ t_5 t_3) t_7)
       (+ (+ (+ (- 1.0 (sqrt x)) t_4) t_1) t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = fma(-0.125, sqrt((1.0 / z)), (0.5 * sqrt(z))) / z;
	double t_4 = sqrt((y + 1.0)) - sqrt(y);
	double t_5 = (sqrt((x + 1.0)) - sqrt(x)) + t_4;
	double t_6 = (t_5 + t_1) + t_2;
	double t_7 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (t_6 <= 4e-6) {
		tmp = (((0.5 * sqrt((1.0 / x))) + (0.5 * sqrt((1.0 / y)))) + t_3) + t_7;
	} else if (t_6 <= 2.002) {
		tmp = (t_5 + t_3) + t_7;
	} else {
		tmp = (((1.0 - sqrt(x)) + t_4) + t_1) + t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = Float64(fma(-0.125, sqrt(Float64(1.0 / z)), Float64(0.5 * sqrt(z))) / z)
	t_4 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_5 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_4)
	t_6 = Float64(Float64(t_5 + t_1) + t_2)
	t_7 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (t_6 <= 4e-6)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + t_3) + t_7);
	elseif (t_6 <= 2.002)
		tmp = Float64(Float64(t_5 + t_3) + t_7);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_4) + t_1) + t_2);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.125 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 4e-6], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$7), $MachinePrecision], If[LessEqual[t$95$6, 2.002], N[(N[(t$95$5 + t$95$3), $MachinePrecision] + t$95$7), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.99999999999999982e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 29.5

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

   10. Applied rewrites 29.5%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 10.7

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

   13. Applied rewrites 10.7%

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

   if 3.99999999999999982e-6 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.0019999999999998

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if 2.0019999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

Alternative 3: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{y}}\right) + \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{z}}, 0.5 \cdot \sqrt{z}\right)}{z}\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           (- (sqrt (+ z 1.0)) (sqrt z)))
          (- (sqrt (+ t 1.0)) (sqrt t)))))
   (if (<= t_1 4e-6)
     (+
      (+
       (+ (* 0.5 (sqrt (/ 1.0 x))) (* 0.5 (sqrt (/ 1.0 y))))
       (/ (fma -0.125 (sqrt (/ 1.0 z)) (* 0.5 (sqrt z))) z))
      (* 0.5 (sqrt (/ 1.0 t))))
     t_1)))

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))) + (sqrt((t + 1.0)) - sqrt(t));
	double tmp;
	if (t_1 <= 4e-6) {
		tmp = (((0.5 * sqrt((1.0 / x))) + (0.5 * sqrt((1.0 / y)))) + (fma(-0.125, sqrt((1.0 / z)), (0.5 * sqrt(z))) / z)) + (0.5 * sqrt((1.0 / t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
	tmp = 0.0
	if (t_1 <= 4e-6)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + Float64(0.5 * sqrt(Float64(1.0 / y)))) + Float64(fma(-0.125, sqrt(Float64(1.0 / z)), Float64(0.5 * sqrt(z))) / z)) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	else
		tmp = 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[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-6], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.99999999999999982e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 63.0

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 29.5

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

   10. Applied rewrites 29.5%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 10.7

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

   13. Applied rewrites 10.7%

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

   if 3.99999999999999982e-6 < (+.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 91.6%

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

Alternative 4: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1} - \sqrt{y}\\ t_4 := \left(\sqrt{x + 1} - \sqrt{x}\right) + t\_3\\ t_5 := \left(t\_4 + t\_1\right) + t\_2\\ t_6 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{if}\;t\_5 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{x}} + t\_3\right) + t\_1\right) + t\_6\\ \mathbf{elif}\;t\_5 \leq 2.0001:\\ \;\;\;\;\left(t\_4 + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_4 (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_3))
        (t_5 (+ (+ t_4 t_1) t_2))
        (t_6 (* 0.5 (sqrt (/ 1.0 t)))))
   (if (<= t_5 4e-6)
     (+ (+ (+ (* 0.5 (sqrt (/ 1.0 x))) t_3) t_1) t_6)
     (if (<= t_5 2.0001)
       (+ (+ t_4 (* 0.5 (sqrt (/ 1.0 z)))) t_6)
       (+
        (-
         (+ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt (+ 1.0 z))))
         (+ (sqrt x) (+ (sqrt y) (sqrt z))))
        t_2)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((y + 1.0)) - sqrt(y);
	double t_4 = (sqrt((x + 1.0)) - sqrt(x)) + t_3;
	double t_5 = (t_4 + t_1) + t_2;
	double t_6 = 0.5 * sqrt((1.0 / t));
	double tmp;
	if (t_5 <= 4e-6) {
		tmp = (((0.5 * sqrt((1.0 / x))) + t_3) + t_1) + t_6;
	} else if (t_5 <= 2.0001) {
		tmp = (t_4 + (0.5 * sqrt((1.0 / z)))) + t_6;
	} else {
		tmp = ((1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))) + t_2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = sqrt((y + 1.0d0)) - sqrt(y)
    t_4 = (sqrt((x + 1.0d0)) - sqrt(x)) + t_3
    t_5 = (t_4 + t_1) + t_2
    t_6 = 0.5d0 * sqrt((1.0d0 / t))
    if (t_5 <= 4d-6) then
        tmp = (((0.5d0 * sqrt((1.0d0 / x))) + t_3) + t_1) + t_6
    else if (t_5 <= 2.0001d0) then
        tmp = (t_4 + (0.5d0 * sqrt((1.0d0 / z)))) + t_6
    else
        tmp = ((1.0d0 + (sqrt((1.0d0 + y)) + sqrt((1.0d0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))) + t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_4 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) + t_3;
	double t_5 = (t_4 + t_1) + t_2;
	double t_6 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (t_5 <= 4e-6) {
		tmp = (((0.5 * Math.sqrt((1.0 / x))) + t_3) + t_1) + t_6;
	} else if (t_5 <= 2.0001) {
		tmp = (t_4 + (0.5 * Math.sqrt((1.0 / z)))) + t_6;
	} else {
		tmp = ((1.0 + (Math.sqrt((1.0 + y)) + Math.sqrt((1.0 + z)))) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)))) + t_2;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_4 = (math.sqrt((x + 1.0)) - math.sqrt(x)) + t_3
	t_5 = (t_4 + t_1) + t_2
	t_6 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if t_5 <= 4e-6:
		tmp = (((0.5 * math.sqrt((1.0 / x))) + t_3) + t_1) + t_6
	elif t_5 <= 2.0001:
		tmp = (t_4 + (0.5 * math.sqrt((1.0 / z)))) + t_6
	else:
		tmp = ((1.0 + (math.sqrt((1.0 + y)) + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))) + t_2
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_4 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_3)
	t_5 = Float64(Float64(t_4 + t_1) + t_2)
	t_6 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	tmp = 0.0
	if (t_5 <= 4e-6)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_3) + t_1) + t_6);
	elseif (t_5 <= 2.0001)
		tmp = Float64(Float64(t_4 + Float64(0.5 * sqrt(Float64(1.0 / z)))) + t_6);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + z)))) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))) + t_2);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = sqrt((y + 1.0)) - sqrt(y);
	t_4 = (sqrt((x + 1.0)) - sqrt(x)) + t_3;
	t_5 = (t_4 + t_1) + t_2;
	t_6 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (t_5 <= 4e-6)
		tmp = (((0.5 * sqrt((1.0 / x))) + t_3) + t_1) + t_6;
	elseif (t_5 <= 2.0001)
		tmp = (t_4 + (0.5 * sqrt((1.0 / z)))) + t_6;
	else
		tmp = ((1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 4e-6], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 2.0001], N[(N[(t$95$4 + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.99999999999999982e-6

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 10.5

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

   10. Applied rewrites 10.5%

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

   if 3.99999999999999982e-6 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00010000000000021

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 64.8

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 64.9

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

   7. Applied rewrites 64.9%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 38.4

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

   4. Applied rewrites 38.4%

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

Alternative 5: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;x \leq 360000000:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + t\_1\right) + t\_2\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{x}} + t\_1\right) + t\_2\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

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 (+ y 1.0)) (sqrt y)))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z))))
   (if (<= x 360000000.0)
     (+
      (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) t_1) t_2)
      (- (sqrt (+ t 1.0)) (sqrt t)))
     (+ (+ (+ (* 0.5 (sqrt (/ 1.0 x))) t_1) t_2) (* 0.5 (sqrt (/ 1.0 t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double t_2 = sqrt((z + 1.0)) - sqrt(z);
	double tmp;
	if (x <= 360000000.0) {
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + t_1) + t_2) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (((0.5 * sqrt((1.0 / x))) + t_1) + t_2) + (0.5 * sqrt((1.0 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0)) - sqrt(y)
    t_2 = sqrt((z + 1.0d0)) - sqrt(z)
    if (x <= 360000000.0d0) then
        tmp = (((sqrt((x + 1.0d0)) - sqrt(x)) + t_1) + t_2) + (sqrt((t + 1.0d0)) - sqrt(t))
    else
        tmp = (((0.5d0 * sqrt((1.0d0 / x))) + t_1) + t_2) + (0.5d0 * sqrt((1.0d0 / 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 t_1 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_2 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double tmp;
	if (x <= 360000000.0) {
		tmp = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + t_1) + t_2) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else {
		tmp = (((0.5 * Math.sqrt((1.0 / x))) + t_1) + t_2) + (0.5 * Math.sqrt((1.0 / t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_2 = math.sqrt((z + 1.0)) - math.sqrt(z)
	tmp = 0
	if x <= 360000000.0:
		tmp = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + t_1) + t_2) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	else:
		tmp = (((0.5 * math.sqrt((1.0 / x))) + t_1) + t_2) + (0.5 * math.sqrt((1.0 / t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	tmp = 0.0
	if (x <= 360000000.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + t_1) + t_2) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_1) + t_2) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0)) - sqrt(y);
	t_2 = sqrt((z + 1.0)) - sqrt(z);
	tmp = 0.0;
	if (x <= 360000000.0)
		tmp = (((sqrt((x + 1.0)) - sqrt(x)) + t_1) + t_2) + (sqrt((t + 1.0)) - sqrt(t));
	else
		tmp = (((0.5 * sqrt((1.0 / x))) + t_1) + t_2) + (0.5 * sqrt((1.0 / 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_] := Block[{t$95$1 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 360000000.0], N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6e8

   1. Initial program 91.6%

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

   if 3.6e8 < x

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 10.5

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

   10. Applied rewrites 10.5%

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

Alternative 6: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{y + 1} - \sqrt{y}\\ \mathbf{if}\;x \leq 0.52:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_2\right) + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{x}} + t\_2\right) + t\_1\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ y 1.0)) (sqrt y))))
   (if (<= x 0.52)
     (+ (+ (+ (- 1.0 (sqrt x)) t_2) t_1) (- (sqrt (+ t 1.0)) (sqrt t)))
     (+ (+ (+ (* 0.5 (sqrt (/ 1.0 x))) t_2) t_1) (* 0.5 (sqrt (/ 1.0 t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((y + 1.0)) - sqrt(y);
	double tmp;
	if (x <= 0.52) {
		tmp = (((1.0 - sqrt(x)) + t_2) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	} else {
		tmp = (((0.5 * sqrt((1.0 / x))) + t_2) + t_1) + (0.5 * sqrt((1.0 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((y + 1.0d0)) - sqrt(y)
    if (x <= 0.52d0) then
        tmp = (((1.0d0 - sqrt(x)) + t_2) + t_1) + (sqrt((t + 1.0d0)) - sqrt(t))
    else
        tmp = (((0.5d0 * sqrt((1.0d0 / x))) + t_2) + t_1) + (0.5d0 * sqrt((1.0d0 / 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 t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double tmp;
	if (x <= 0.52) {
		tmp = (((1.0 - Math.sqrt(x)) + t_2) + t_1) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
	} else {
		tmp = (((0.5 * Math.sqrt((1.0 / x))) + t_2) + t_1) + (0.5 * Math.sqrt((1.0 / t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((y + 1.0)) - math.sqrt(y)
	tmp = 0
	if x <= 0.52:
		tmp = (((1.0 - math.sqrt(x)) + t_2) + t_1) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	else:
		tmp = (((0.5 * math.sqrt((1.0 / x))) + t_2) + t_1) + (0.5 * math.sqrt((1.0 / t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	tmp = 0.0
	if (x <= 0.52)
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_2) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) + t_2) + t_1) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((y + 1.0)) - sqrt(y);
	tmp = 0.0;
	if (x <= 0.52)
		tmp = (((1.0 - sqrt(x)) + t_2) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	else
		tmp = (((0.5 * sqrt((1.0 / x))) + t_2) + t_1) + (0.5 * sqrt((1.0 / 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_] := 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[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.52], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.52000000000000002

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   if 0.52000000000000002 < x

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 10.5

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

   10. Applied rewrites 10.5%

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

Alternative 7: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := 1 - \sqrt{y}\\ t_2 := 1 - \sqrt{x}\\ t_3 := \sqrt{z + 1} - \sqrt{z}\\ t_4 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ t_5 := \sqrt{y + 1} - \sqrt{y}\\ t_6 := \sqrt{x + 1} - \sqrt{x}\\ t_7 := \sqrt{t + 1} - \sqrt{t}\\ t_8 := \left(\left(t\_6 + t\_5\right) + t\_3\right) + t\_7\\ \mathbf{if}\;t\_8 \leq 2.0001:\\ \;\;\;\;\left(\left(t\_2 + t\_5\right) + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_4\\ \mathbf{elif}\;t\_8 \leq 3.00005:\\ \;\;\;\;\left(\left(t\_2 + t\_1\right) + t\_3\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_6 + t\_1\right) + \left(1 - \sqrt{z}\right)\right) + t\_7\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 1.0 (sqrt y)))
        (t_2 (- 1.0 (sqrt x)))
        (t_3 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_4 (* 0.5 (sqrt (/ 1.0 t))))
        (t_5 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_6 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_7 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_8 (+ (+ (+ t_6 t_5) t_3) t_7)))
   (if (<= t_8 2.0001)
     (+ (+ (+ t_2 t_5) (* 0.5 (sqrt (/ 1.0 z)))) t_4)
     (if (<= t_8 3.00005)
       (+ (+ (+ t_2 t_1) t_3) t_4)
       (+ (+ (+ t_6 t_1) (- 1.0 (sqrt z))) t_7)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - sqrt(y);
	double t_2 = 1.0 - sqrt(x);
	double t_3 = sqrt((z + 1.0)) - sqrt(z);
	double t_4 = 0.5 * sqrt((1.0 / t));
	double t_5 = sqrt((y + 1.0)) - sqrt(y);
	double t_6 = sqrt((x + 1.0)) - sqrt(x);
	double t_7 = sqrt((t + 1.0)) - sqrt(t);
	double t_8 = ((t_6 + t_5) + t_3) + t_7;
	double tmp;
	if (t_8 <= 2.0001) {
		tmp = ((t_2 + t_5) + (0.5 * sqrt((1.0 / z)))) + t_4;
	} else if (t_8 <= 3.00005) {
		tmp = ((t_2 + t_1) + t_3) + t_4;
	} else {
		tmp = ((t_6 + t_1) + (1.0 - sqrt(z))) + t_7;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = 1.0d0 - sqrt(y)
    t_2 = 1.0d0 - sqrt(x)
    t_3 = sqrt((z + 1.0d0)) - sqrt(z)
    t_4 = 0.5d0 * sqrt((1.0d0 / t))
    t_5 = sqrt((y + 1.0d0)) - sqrt(y)
    t_6 = sqrt((x + 1.0d0)) - sqrt(x)
    t_7 = sqrt((t + 1.0d0)) - sqrt(t)
    t_8 = ((t_6 + t_5) + t_3) + t_7
    if (t_8 <= 2.0001d0) then
        tmp = ((t_2 + t_5) + (0.5d0 * sqrt((1.0d0 / z)))) + t_4
    else if (t_8 <= 3.00005d0) then
        tmp = ((t_2 + t_1) + t_3) + t_4
    else
        tmp = ((t_6 + t_1) + (1.0d0 - sqrt(z))) + t_7
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 - Math.sqrt(y);
	double t_2 = 1.0 - Math.sqrt(x);
	double t_3 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_4 = 0.5 * Math.sqrt((1.0 / t));
	double t_5 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_6 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_7 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_8 = ((t_6 + t_5) + t_3) + t_7;
	double tmp;
	if (t_8 <= 2.0001) {
		tmp = ((t_2 + t_5) + (0.5 * Math.sqrt((1.0 / z)))) + t_4;
	} else if (t_8 <= 3.00005) {
		tmp = ((t_2 + t_1) + t_3) + t_4;
	} else {
		tmp = ((t_6 + t_1) + (1.0 - Math.sqrt(z))) + t_7;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = 1.0 - math.sqrt(y)
	t_2 = 1.0 - math.sqrt(x)
	t_3 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_4 = 0.5 * math.sqrt((1.0 / t))
	t_5 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_6 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_7 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_8 = ((t_6 + t_5) + t_3) + t_7
	tmp = 0
	if t_8 <= 2.0001:
		tmp = ((t_2 + t_5) + (0.5 * math.sqrt((1.0 / z)))) + t_4
	elif t_8 <= 3.00005:
		tmp = ((t_2 + t_1) + t_3) + t_4
	else:
		tmp = ((t_6 + t_1) + (1.0 - math.sqrt(z))) + t_7
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(1.0 - sqrt(y))
	t_2 = Float64(1.0 - sqrt(x))
	t_3 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_4 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	t_5 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_6 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_7 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_8 = Float64(Float64(Float64(t_6 + t_5) + t_3) + t_7)
	tmp = 0.0
	if (t_8 <= 2.0001)
		tmp = Float64(Float64(Float64(t_2 + t_5) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + t_4);
	elseif (t_8 <= 3.00005)
		tmp = Float64(Float64(Float64(t_2 + t_1) + t_3) + t_4);
	else
		tmp = Float64(Float64(Float64(t_6 + t_1) + Float64(1.0 - sqrt(z))) + t_7);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 - sqrt(y);
	t_2 = 1.0 - sqrt(x);
	t_3 = sqrt((z + 1.0)) - sqrt(z);
	t_4 = 0.5 * sqrt((1.0 / t));
	t_5 = sqrt((y + 1.0)) - sqrt(y);
	t_6 = sqrt((x + 1.0)) - sqrt(x);
	t_7 = sqrt((t + 1.0)) - sqrt(t);
	t_8 = ((t_6 + t_5) + t_3) + t_7;
	tmp = 0.0;
	if (t_8 <= 2.0001)
		tmp = ((t_2 + t_5) + (0.5 * sqrt((1.0 / z)))) + t_4;
	elseif (t_8 <= 3.00005)
		tmp = ((t_2 + t_1) + t_3) + t_4;
	else
		tmp = ((t_6 + t_1) + (1.0 - sqrt(z))) + t_7;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[(t$95$6 + t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$7), $MachinePrecision]}, If[LessEqual[t$95$8, 2.0001], N[(N[(N[(t$95$2 + t$95$5), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$8, 3.00005], N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision], N[(N[(N[(t$95$6 + t$95$1), $MachinePrecision] + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$7), $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))) < 2.00010000000000021

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 63.3

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

   10. Applied rewrites 63.3%

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

   if 2.00010000000000021 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.0000499999999999

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 60.5

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

   10. Applied rewrites 60.5%

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

   if 3.0000499999999999 < (+.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 91.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 65.8

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 29.6

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

   7. Applied rewrites 29.6%

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

Alternative 8: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ t_3 := \sqrt{z + 1} - \sqrt{z}\\ t_4 := 1 - \sqrt{y}\\ t_5 := \sqrt{x + 1} - \sqrt{x}\\ t_6 := \sqrt{t + 1} - \sqrt{t}\\ t_7 := \left(\left(t\_5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_3\right) + t\_6\\ \mathbf{if}\;t\_7 \leq 1:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{x}\right) + t\_3\right) + t\_2\\ \mathbf{elif}\;t\_7 \leq 1.99999999998:\\ \;\;\;\;\left(\left(t\_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + t\_6\\ \mathbf{elif}\;t\_7 \leq 3.00005:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_4\right) + t\_3\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_5 + t\_4\right) + \left(1 - \sqrt{z}\right)\right) + t\_6\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (* 0.5 (sqrt (/ 1.0 t))))
        (t_3 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_4 (- 1.0 (sqrt y)))
        (t_5 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_6 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_7 (+ (+ (+ t_5 (- (sqrt (+ y 1.0)) (sqrt y))) t_3) t_6)))
   (if (<= t_7 1.0)
     (+ (+ (- t_1 (sqrt x)) t_3) t_2)
     (if (<= t_7 1.99999999998)
       (+ (- (+ t_1 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y))) t_6)
       (if (<= t_7 3.00005)
         (+ (+ (+ (- 1.0 (sqrt x)) t_4) t_3) t_2)
         (+ (+ (+ t_5 t_4) (- 1.0 (sqrt z))) t_6))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = 0.5 * sqrt((1.0 / t));
	double t_3 = sqrt((z + 1.0)) - sqrt(z);
	double t_4 = 1.0 - sqrt(y);
	double t_5 = sqrt((x + 1.0)) - sqrt(x);
	double t_6 = sqrt((t + 1.0)) - sqrt(t);
	double t_7 = ((t_5 + (sqrt((y + 1.0)) - sqrt(y))) + t_3) + t_6;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = ((t_1 - sqrt(x)) + t_3) + t_2;
	} else if (t_7 <= 1.99999999998) {
		tmp = ((t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y))) + t_6;
	} else if (t_7 <= 3.00005) {
		tmp = (((1.0 - sqrt(x)) + t_4) + t_3) + t_2;
	} else {
		tmp = ((t_5 + t_4) + (1.0 - sqrt(z))) + t_6;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = 0.5d0 * sqrt((1.0d0 / t))
    t_3 = sqrt((z + 1.0d0)) - sqrt(z)
    t_4 = 1.0d0 - sqrt(y)
    t_5 = sqrt((x + 1.0d0)) - sqrt(x)
    t_6 = sqrt((t + 1.0d0)) - sqrt(t)
    t_7 = ((t_5 + (sqrt((y + 1.0d0)) - sqrt(y))) + t_3) + t_6
    if (t_7 <= 1.0d0) then
        tmp = ((t_1 - sqrt(x)) + t_3) + t_2
    else if (t_7 <= 1.99999999998d0) then
        tmp = ((t_1 + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y))) + t_6
    else if (t_7 <= 3.00005d0) then
        tmp = (((1.0d0 - sqrt(x)) + t_4) + t_3) + t_2
    else
        tmp = ((t_5 + t_4) + (1.0d0 - sqrt(z))) + t_6
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = 0.5 * Math.sqrt((1.0 / t));
	double t_3 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_4 = 1.0 - Math.sqrt(y);
	double t_5 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_6 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_7 = ((t_5 + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_3) + t_6;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = ((t_1 - Math.sqrt(x)) + t_3) + t_2;
	} else if (t_7 <= 1.99999999998) {
		tmp = ((t_1 + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y))) + t_6;
	} else if (t_7 <= 3.00005) {
		tmp = (((1.0 - Math.sqrt(x)) + t_4) + t_3) + t_2;
	} else {
		tmp = ((t_5 + t_4) + (1.0 - Math.sqrt(z))) + t_6;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = 0.5 * math.sqrt((1.0 / t))
	t_3 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_4 = 1.0 - math.sqrt(y)
	t_5 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_6 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_7 = ((t_5 + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_3) + t_6
	tmp = 0
	if t_7 <= 1.0:
		tmp = ((t_1 - math.sqrt(x)) + t_3) + t_2
	elif t_7 <= 1.99999999998:
		tmp = ((t_1 + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y))) + t_6
	elif t_7 <= 3.00005:
		tmp = (((1.0 - math.sqrt(x)) + t_4) + t_3) + t_2
	else:
		tmp = ((t_5 + t_4) + (1.0 - math.sqrt(z))) + t_6
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	t_3 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_4 = Float64(1.0 - sqrt(y))
	t_5 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_6 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_7 = Float64(Float64(Float64(t_5 + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_3) + t_6)
	tmp = 0.0
	if (t_7 <= 1.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(x)) + t_3) + t_2);
	elseif (t_7 <= 1.99999999998)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y))) + t_6);
	elseif (t_7 <= 3.00005)
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_4) + t_3) + t_2);
	else
		tmp = Float64(Float64(Float64(t_5 + t_4) + Float64(1.0 - sqrt(z))) + t_6);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = 0.5 * sqrt((1.0 / t));
	t_3 = sqrt((z + 1.0)) - sqrt(z);
	t_4 = 1.0 - sqrt(y);
	t_5 = sqrt((x + 1.0)) - sqrt(x);
	t_6 = sqrt((t + 1.0)) - sqrt(t);
	t_7 = ((t_5 + (sqrt((y + 1.0)) - sqrt(y))) + t_3) + t_6;
	tmp = 0.0;
	if (t_7 <= 1.0)
		tmp = ((t_1 - sqrt(x)) + t_3) + t_2;
	elseif (t_7 <= 1.99999999998)
		tmp = ((t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y))) + t_6;
	elseif (t_7 <= 3.00005)
		tmp = (((1.0 - sqrt(x)) + t_4) + t_3) + t_2;
	else
		tmp = ((t_5 + t_4) + (1.0 - sqrt(z))) + t_6;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(t$95$5 + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$6), $MachinePrecision]}, If[LessEqual[t$95$7, 1.0], N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$7, 1.99999999998], N[(N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision], If[LessEqual[t$95$7, 3.00005], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(t$95$5 + t$95$4), $MachinePrecision] + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 36.1

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

   7. Applied rewrites 36.1%

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

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.99999999998

   1. Initial program 91.6%

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

   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. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 47.8

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

   4. Applied rewrites 47.8%

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

   if 1.99999999998 < (+.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))) < 3.0000499999999999

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 60.5

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

   10. Applied rewrites 60.5%

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

   if 3.0000499999999999 < (+.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 91.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 65.8

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 29.6

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

   7. Applied rewrites 29.6%

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

Alternative 9: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := 0.5 \cdot \sqrt{\frac{1}{t}}\\ t_3 := \sqrt{z + 1} - \sqrt{z}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_3\right) + t\_4\\ \mathbf{if}\;t\_5 \leq 1:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{x}\right) + t\_3\right) + t\_2\\ \mathbf{elif}\;t\_5 \leq 1.99999999998:\\ \;\;\;\;\left(\left(t\_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 3.5:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \sqrt{y}\right)\right) + t\_3\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \left(t\_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (* 0.5 (sqrt (/ 1.0 t))))
        (t_3 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_3)
          t_4)))
   (if (<= t_5 1.0)
     (+ (+ (- t_1 (sqrt x)) t_3) t_2)
     (if (<= t_5 1.99999999998)
       (+ (- (+ t_1 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y))) t_4)
       (if (<= t_5 3.5)
         (+ (+ (+ (- 1.0 (sqrt x)) (- 1.0 (sqrt y))) t_3) t_2)
         (-
          (+ 2.0 (+ t_1 (sqrt (+ 1.0 z))))
          (+ (sqrt t) (+ (sqrt x) (+ (sqrt y) (sqrt z))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = 0.5 * sqrt((1.0 / t));
	double t_3 = sqrt((z + 1.0)) - sqrt(z);
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_3) + t_4;
	double tmp;
	if (t_5 <= 1.0) {
		tmp = ((t_1 - sqrt(x)) + t_3) + t_2;
	} else if (t_5 <= 1.99999999998) {
		tmp = ((t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y))) + t_4;
	} else if (t_5 <= 3.5) {
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_3) + t_2;
	} else {
		tmp = (2.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = 0.5d0 * sqrt((1.0d0 / t))
    t_3 = sqrt((z + 1.0d0)) - sqrt(z)
    t_4 = sqrt((t + 1.0d0)) - sqrt(t)
    t_5 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_3) + t_4
    if (t_5 <= 1.0d0) then
        tmp = ((t_1 - sqrt(x)) + t_3) + t_2
    else if (t_5 <= 1.99999999998d0) then
        tmp = ((t_1 + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y))) + t_4
    else if (t_5 <= 3.5d0) then
        tmp = (((1.0d0 - sqrt(x)) + (1.0d0 - sqrt(y))) + t_3) + t_2
    else
        tmp = (2.0d0 + (t_1 + sqrt((1.0d0 + z)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = 0.5 * Math.sqrt((1.0 / t));
	double t_3 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_4 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_5 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_3) + t_4;
	double tmp;
	if (t_5 <= 1.0) {
		tmp = ((t_1 - Math.sqrt(x)) + t_3) + t_2;
	} else if (t_5 <= 1.99999999998) {
		tmp = ((t_1 + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y))) + t_4;
	} else if (t_5 <= 3.5) {
		tmp = (((1.0 - Math.sqrt(x)) + (1.0 - Math.sqrt(y))) + t_3) + t_2;
	} else {
		tmp = (2.0 + (t_1 + Math.sqrt((1.0 + z)))) - (Math.sqrt(t) + (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = 0.5 * math.sqrt((1.0 / t))
	t_3 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_4 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_5 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_3) + t_4
	tmp = 0
	if t_5 <= 1.0:
		tmp = ((t_1 - math.sqrt(x)) + t_3) + t_2
	elif t_5 <= 1.99999999998:
		tmp = ((t_1 + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y))) + t_4
	elif t_5 <= 3.5:
		tmp = (((1.0 - math.sqrt(x)) + (1.0 - math.sqrt(y))) + t_3) + t_2
	else:
		tmp = (2.0 + (t_1 + math.sqrt((1.0 + z)))) - (math.sqrt(t) + (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(0.5 * sqrt(Float64(1.0 / t)))
	t_3 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_3) + t_4)
	tmp = 0.0
	if (t_5 <= 1.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(x)) + t_3) + t_2);
	elseif (t_5 <= 1.99999999998)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y))) + t_4);
	elseif (t_5 <= 3.5)
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(1.0 - sqrt(y))) + t_3) + t_2);
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 + sqrt(Float64(1.0 + z)))) - Float64(sqrt(t) + Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = 0.5 * sqrt((1.0 / t));
	t_3 = sqrt((z + 1.0)) - sqrt(z);
	t_4 = sqrt((t + 1.0)) - sqrt(t);
	t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_3) + t_4;
	tmp = 0.0;
	if (t_5 <= 1.0)
		tmp = ((t_1 - sqrt(x)) + t_3) + t_2;
	elseif (t_5 <= 1.99999999998)
		tmp = ((t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y))) + t_4;
	elseif (t_5 <= 3.5)
		tmp = (((1.0 - sqrt(x)) + (1.0 - sqrt(y))) + t_3) + t_2;
	else
		tmp = (2.0 + (t_1 + sqrt((1.0 + z)))) - (sqrt(t) + (sqrt(x) + (sqrt(y) + sqrt(z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $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[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0], N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 1.99999999998], N[(N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 3.5], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $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))) < 1

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 36.1

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

   7. Applied rewrites 36.1%

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

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.99999999998

   1. Initial program 91.6%

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

   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. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 47.8

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

   4. Applied rewrites 47.8%

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

   if 1.99999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.5

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 60.5

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

   10. Applied rewrites 60.5%

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

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

   1. Initial program 91.6%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 6.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 6.8

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

   7. Applied rewrites 6.8%

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

Alternative 10: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{y + 1} - \sqrt{y}\\ t_3 := \sqrt{x + 1} - \sqrt{x}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;\left(\left(t\_3 + t\_2\right) + t\_1\right) + t\_4 \leq 3.00005:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + t\_2\right) + t\_1\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_3 + \left(1 - \sqrt{y}\right)\right) + \left(1 - \sqrt{z}\right)\right) + t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_2 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_3 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= (+ (+ (+ t_3 t_2) t_1) t_4) 3.00005)
     (+ (+ (+ (- 1.0 (sqrt x)) t_2) t_1) (* 0.5 (sqrt (/ 1.0 t))))
     (+ (+ (+ t_3 (- 1.0 (sqrt y))) (- 1.0 (sqrt z))) t_4))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((y + 1.0)) - sqrt(y);
	double t_3 = sqrt((x + 1.0)) - sqrt(x);
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if ((((t_3 + t_2) + t_1) + t_4) <= 3.00005) {
		tmp = (((1.0 - sqrt(x)) + t_2) + t_1) + (0.5 * sqrt((1.0 / t)));
	} else {
		tmp = ((t_3 + (1.0 - sqrt(y))) + (1.0 - sqrt(z))) + t_4;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = sqrt((y + 1.0d0)) - sqrt(y)
    t_3 = sqrt((x + 1.0d0)) - sqrt(x)
    t_4 = sqrt((t + 1.0d0)) - sqrt(t)
    if ((((t_3 + t_2) + t_1) + t_4) <= 3.00005d0) then
        tmp = (((1.0d0 - sqrt(x)) + t_2) + t_1) + (0.5d0 * sqrt((1.0d0 / t)))
    else
        tmp = ((t_3 + (1.0d0 - sqrt(y))) + (1.0d0 - sqrt(z))) + t_4
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = Math.sqrt((y + 1.0)) - Math.sqrt(y);
	double t_3 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_4 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if ((((t_3 + t_2) + t_1) + t_4) <= 3.00005) {
		tmp = (((1.0 - Math.sqrt(x)) + t_2) + t_1) + (0.5 * Math.sqrt((1.0 / t)));
	} else {
		tmp = ((t_3 + (1.0 - Math.sqrt(y))) + (1.0 - Math.sqrt(z))) + t_4;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = math.sqrt((y + 1.0)) - math.sqrt(y)
	t_3 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_4 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if (((t_3 + t_2) + t_1) + t_4) <= 3.00005:
		tmp = (((1.0 - math.sqrt(x)) + t_2) + t_1) + (0.5 * math.sqrt((1.0 / t)))
	else:
		tmp = ((t_3 + (1.0 - math.sqrt(y))) + (1.0 - math.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(Float64(z + 1.0)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_3 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 + t_2) + t_1) + t_4) <= 3.00005)
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + t_2) + t_1) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	else
		tmp = Float64(Float64(Float64(t_3 + Float64(1.0 - sqrt(y))) + Float64(1.0 - sqrt(z))) + t_4);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = sqrt((y + 1.0)) - sqrt(y);
	t_3 = sqrt((x + 1.0)) - sqrt(x);
	t_4 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if ((((t_3 + t_2) + t_1) + t_4) <= 3.00005)
		tmp = (((1.0 - sqrt(x)) + t_2) + t_1) + (0.5 * sqrt((1.0 / t)));
	else
		tmp = ((t_3 + (1.0 - sqrt(y))) + (1.0 - sqrt(z))) + t_4;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], 3.00005], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 + N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.0000499999999999

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if 3.0000499999999999 < (+.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 91.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 65.8

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lift-sqrt.f64 29.6

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

   7. Applied rewrites 29.6%

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

Alternative 11: 90.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(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 = (((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (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[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 90.3%

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

Alternative 12: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1} - \sqrt{z}\\ t_3 := \sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ t_5 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + t\_2\right) + t\_4\\ t_6 := \sqrt{1 + z}\\ \mathbf{if}\;t\_5 \leq 1:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{x}\right) + t\_2\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\left(\left(t\_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 3.5:\\ \;\;\;\;\left(t\_1 + \left(1 + t\_6\right)\right) - t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \left(t\_1 + t\_6\right)\right) - \left(\sqrt{t} + t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_3 (+ (sqrt x) (+ (sqrt y) (sqrt z))))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_5
         (+
          (+
           (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
           t_2)
          t_4))
        (t_6 (sqrt (+ 1.0 z))))
   (if (<= t_5 1.0)
     (+ (+ (- t_1 (sqrt x)) t_2) (* 0.5 (sqrt (/ 1.0 t))))
     (if (<= t_5 2.0)
       (+ (- (+ t_1 (sqrt (+ 1.0 y))) (+ (sqrt x) (sqrt y))) t_4)
       (if (<= t_5 3.5)
         (- (+ t_1 (+ 1.0 t_6)) t_3)
         (- (+ 2.0 (+ t_1 t_6)) (+ (sqrt t) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((z + 1.0)) - sqrt(z);
	double t_3 = sqrt(x) + (sqrt(y) + sqrt(z));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2) + t_4;
	double t_6 = sqrt((1.0 + z));
	double tmp;
	if (t_5 <= 1.0) {
		tmp = ((t_1 - sqrt(x)) + t_2) + (0.5 * sqrt((1.0 / t)));
	} else if (t_5 <= 2.0) {
		tmp = ((t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y))) + t_4;
	} else if (t_5 <= 3.5) {
		tmp = (t_1 + (1.0 + t_6)) - t_3;
	} else {
		tmp = (2.0 + (t_1 + t_6)) - (sqrt(t) + t_3);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((z + 1.0d0)) - sqrt(z)
    t_3 = sqrt(x) + (sqrt(y) + sqrt(z))
    t_4 = sqrt((t + 1.0d0)) - sqrt(t)
    t_5 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_2) + t_4
    t_6 = sqrt((1.0d0 + z))
    if (t_5 <= 1.0d0) then
        tmp = ((t_1 - sqrt(x)) + t_2) + (0.5d0 * sqrt((1.0d0 / t)))
    else if (t_5 <= 2.0d0) then
        tmp = ((t_1 + sqrt((1.0d0 + y))) - (sqrt(x) + sqrt(y))) + t_4
    else if (t_5 <= 3.5d0) then
        tmp = (t_1 + (1.0d0 + t_6)) - t_3
    else
        tmp = (2.0d0 + (t_1 + t_6)) - (sqrt(t) + t_3)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_3 = Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z));
	double t_4 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_5 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_2) + t_4;
	double t_6 = Math.sqrt((1.0 + z));
	double tmp;
	if (t_5 <= 1.0) {
		tmp = ((t_1 - Math.sqrt(x)) + t_2) + (0.5 * Math.sqrt((1.0 / t)));
	} else if (t_5 <= 2.0) {
		tmp = ((t_1 + Math.sqrt((1.0 + y))) - (Math.sqrt(x) + Math.sqrt(y))) + t_4;
	} else if (t_5 <= 3.5) {
		tmp = (t_1 + (1.0 + t_6)) - t_3;
	} else {
		tmp = (2.0 + (t_1 + t_6)) - (Math.sqrt(t) + t_3);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_3 = math.sqrt(x) + (math.sqrt(y) + math.sqrt(z))
	t_4 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_5 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_2) + t_4
	t_6 = math.sqrt((1.0 + z))
	tmp = 0
	if t_5 <= 1.0:
		tmp = ((t_1 - math.sqrt(x)) + t_2) + (0.5 * math.sqrt((1.0 / t)))
	elif t_5 <= 2.0:
		tmp = ((t_1 + math.sqrt((1.0 + y))) - (math.sqrt(x) + math.sqrt(y))) + t_4
	elif t_5 <= 3.5:
		tmp = (t_1 + (1.0 + t_6)) - t_3
	else:
		tmp = (2.0 + (t_1 + t_6)) - (math.sqrt(t) + t_3)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_3 = Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z)))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_5 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + t_2) + t_4)
	t_6 = sqrt(Float64(1.0 + z))
	tmp = 0.0
	if (t_5 <= 1.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(x)) + t_2) + Float64(0.5 * sqrt(Float64(1.0 / t))));
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(Float64(t_1 + sqrt(Float64(1.0 + y))) - Float64(sqrt(x) + sqrt(y))) + t_4);
	elseif (t_5 <= 3.5)
		tmp = Float64(Float64(t_1 + Float64(1.0 + t_6)) - t_3);
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 + t_6)) - Float64(sqrt(t) + t_3));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((z + 1.0)) - sqrt(z);
	t_3 = sqrt(x) + (sqrt(y) + sqrt(z));
	t_4 = sqrt((t + 1.0)) - sqrt(t);
	t_5 = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_2) + t_4;
	t_6 = sqrt((1.0 + z));
	tmp = 0.0;
	if (t_5 <= 1.0)
		tmp = ((t_1 - sqrt(x)) + t_2) + (0.5 * sqrt((1.0 / t)));
	elseif (t_5 <= 2.0)
		tmp = ((t_1 + sqrt((1.0 + y))) - (sqrt(x) + sqrt(y))) + t_4;
	elseif (t_5 <= 3.5)
		tmp = (t_1 + (1.0 + t_6)) - t_3;
	else
		tmp = (2.0 + (t_1 + t_6)) - (sqrt(t) + t_3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $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[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 1.0], N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(N[(t$95$1 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 3.5], N[(N[(t$95$1 + N[(1.0 + t$95$6), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 + t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 36.1

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

   7. Applied rewrites 36.1%

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

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

   1. Initial program 91.6%

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

   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. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 47.8

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

   4. Applied rewrites 47.8%

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

   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))) < 3.5

   1. Initial program 91.6%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 32.2

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

   7. Applied rewrites 32.2%

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

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

   1. Initial program 91.6%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 6.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 6.8

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

   7. Applied rewrites 6.8%

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

Alternative 13: 86.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1
    t_3 = sqrt((1.0d0 + y))
    t_4 = sqrt((1.0d0 + x))
    t_5 = 0.5d0 * sqrt((1.0d0 / t))
    if (t_2 <= 1.0d0) then
        tmp = ((t_4 - sqrt(x)) + t_1) + t_5
    else if (t_2 <= 2.0d0) then
        tmp = ((t_4 + t_3) - (sqrt(x) + sqrt(y))) + t_5
    else
        tmp = (1.0d0 + (t_3 + sqrt((1.0d0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0)) - Math.sqrt(z);
	double t_2 = ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + t_1;
	double t_3 = Math.sqrt((1.0 + y));
	double t_4 = Math.sqrt((1.0 + x));
	double t_5 = 0.5 * Math.sqrt((1.0 / t));
	double tmp;
	if (t_2 <= 1.0) {
		tmp = ((t_4 - Math.sqrt(x)) + t_1) + t_5;
	} else if (t_2 <= 2.0) {
		tmp = ((t_4 + t_3) - (Math.sqrt(x) + Math.sqrt(y))) + t_5;
	} else {
		tmp = (1.0 + (t_3 + Math.sqrt((1.0 + z)))) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1
	t_3 = math.sqrt((1.0 + y))
	t_4 = math.sqrt((1.0 + x))
	t_5 = 0.5 * math.sqrt((1.0 / t))
	tmp = 0
	if t_2 <= 1.0:
		tmp = ((t_4 - math.sqrt(x)) + t_1) + t_5
	elif t_2 <= 2.0:
		tmp = ((t_4 + t_3) - (math.sqrt(x) + math.sqrt(y))) + t_5
	else:
		tmp = (1.0 + (t_3 + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	return tmp

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	t_3 = sqrt((1.0 + y));
	t_4 = sqrt((1.0 + x));
	t_5 = 0.5 * sqrt((1.0 / t));
	tmp = 0.0;
	if (t_2 <= 1.0)
		tmp = ((t_4 - sqrt(x)) + t_1) + t_5;
	elseif (t_2 <= 2.0)
		tmp = ((t_4 + t_3) - (sqrt(x) + sqrt(y))) + t_5;
	else
		tmp = (1.0 + (t_3 + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 36.1

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

   7. Applied rewrites 36.1%

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

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

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 47.7

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

   10. Applied rewrites 47.7%

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

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

   1. Initial program 91.6%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 33.0

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

   7. Applied rewrites 33.0%

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

Alternative 14: 86.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0)) - sqrt(z)
    t_2 = ((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + t_1
    t_3 = sqrt((1.0d0 + y))
    t_4 = sqrt((1.0d0 + x))
    if (t_2 <= 1.0d0) then
        tmp = ((t_4 - sqrt(x)) + t_1) + (0.5d0 * sqrt((1.0d0 / t)))
    else if (t_2 <= 2.0d0) then
        tmp = ((t_4 + t_3) - (sqrt(x) + sqrt(y))) + (sqrt((t + 1.0d0)) - sqrt(t))
    else
        tmp = (1.0d0 + (t_3 + sqrt((1.0d0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    end if
    code = tmp
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0)) - math.sqrt(z)
	t_2 = ((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + t_1
	t_3 = math.sqrt((1.0 + y))
	t_4 = math.sqrt((1.0 + x))
	tmp = 0
	if t_2 <= 1.0:
		tmp = ((t_4 - math.sqrt(x)) + t_1) + (0.5 * math.sqrt((1.0 / t)))
	elif t_2 <= 2.0:
		tmp = ((t_4 + t_3) - (math.sqrt(x) + math.sqrt(y))) + (math.sqrt((t + 1.0)) - math.sqrt(t))
	else:
		tmp = (1.0 + (t_3 + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	return tmp

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0)) - sqrt(z);
	t_2 = ((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + t_1;
	t_3 = sqrt((1.0 + y));
	t_4 = sqrt((1.0 + x));
	tmp = 0.0;
	if (t_2 <= 1.0)
		tmp = ((t_4 - sqrt(x)) + t_1) + (0.5 * sqrt((1.0 / t)));
	elseif (t_2 <= 2.0)
		tmp = ((t_4 + t_3) - (sqrt(x) + sqrt(y))) + (sqrt((t + 1.0)) - sqrt(t));
	else
		tmp = (1.0 + (t_3 + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 36.1

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

   7. Applied rewrites 36.1%

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

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

   1. Initial program 91.6%

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

   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. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 47.8

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

   4. Applied rewrites 47.8%

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

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

   1. Initial program 91.6%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 33.0

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

   7. Applied rewrites 33.0%

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

Alternative 15: 59.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+32}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + 0.5 \cdot \sqrt{\frac{1}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3e+32)
   (-
    (+ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt (+ 1.0 z))))
    (+ (sqrt x) (+ (sqrt y) (sqrt z))))
   (+
    (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- (sqrt (+ z 1.0)) (sqrt z)))
    (* 0.5 (sqrt (/ 1.0 t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+32) {
		tmp = (1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = ((sqrt((1.0 + x)) - sqrt(x)) + (sqrt((z + 1.0)) - sqrt(z))) + (0.5 * sqrt((1.0 / 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 <= 3d+32) then
        tmp = (1.0d0 + (sqrt((1.0d0 + y)) + sqrt((1.0d0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = ((sqrt((1.0d0 + x)) - sqrt(x)) + (sqrt((z + 1.0d0)) - sqrt(z))) + (0.5d0 * sqrt((1.0d0 / 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 <= 3e+32) {
		tmp = (1.0 + (Math.sqrt((1.0 + y)) + Math.sqrt((1.0 + z)))) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = ((Math.sqrt((1.0 + x)) - Math.sqrt(x)) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (0.5 * Math.sqrt((1.0 / t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= 3e+32:
		tmp = (1.0 + (math.sqrt((1.0 + y)) + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = ((math.sqrt((1.0 + x)) - math.sqrt(x)) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (0.5 * math.sqrt((1.0 / t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3e+32)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + z)))) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(0.5 * sqrt(Float64(1.0 / 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 <= 3e+32)
		tmp = (1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = ((sqrt((1.0 + x)) - sqrt(x)) + (sqrt((z + 1.0)) - sqrt(z))) + (0.5 * sqrt((1.0 / 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, 3e+32], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3e32

   1. Initial program 91.6%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 33.0

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

   7. Applied rewrites 33.0%

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

   if 3e32 < z

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 36.1

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

   7. Applied rewrites 36.1%

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

Alternative 16: 59.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+32}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + x} - \sqrt{x}\right) + -0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3e+32)
   (-
    (+ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt (+ 1.0 z))))
    (+ (sqrt x) (+ (sqrt y) (sqrt z))))
   (+
    (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (* -0.5 (sqrt (/ 1.0 z))))
    (- (sqrt (+ t 1.0)) (sqrt t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3e+32) {
		tmp = (1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	} else {
		tmp = ((sqrt((1.0 + x)) - sqrt(x)) + (-0.5 * sqrt((1.0 / z)))) + (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 <= 3d+32) then
        tmp = (1.0d0 + (sqrt((1.0d0 + y)) + sqrt((1.0d0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)))
    else
        tmp = ((sqrt((1.0d0 + x)) - sqrt(x)) + ((-0.5d0) * sqrt((1.0d0 / z)))) + (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 <= 3e+32) {
		tmp = (1.0 + (Math.sqrt((1.0 + y)) + Math.sqrt((1.0 + z)))) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
	} else {
		tmp = ((Math.sqrt((1.0 + x)) - Math.sqrt(x)) + (-0.5 * Math.sqrt((1.0 / z)))) + (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 <= 3e+32:
		tmp = (1.0 + (math.sqrt((1.0 + y)) + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (math.sqrt(y) + math.sqrt(z)))
	else:
		tmp = ((math.sqrt((1.0 + x)) - math.sqrt(x)) + (-0.5 * math.sqrt((1.0 / z)))) + (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 <= 3e+32)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + z)))) - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(-0.5 * sqrt(Float64(1.0 / z)))) + 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 <= 3e+32)
		tmp = (1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (sqrt(y) + sqrt(z)));
	else
		tmp = ((sqrt((1.0 + x)) - sqrt(x)) + (-0.5 * sqrt((1.0 / z)))) + (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, 3e+32], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $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 < 3e32

   1. Initial program 91.6%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 33.0

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

   7. Applied rewrites 33.0%

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

   if 3e32 < z

   1. Initial program 91.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-sqrt.f64 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in z around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 30.8

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

   7. Applied rewrites 30.8%

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

Alternative 17: 33.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (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 = (1.0d0 + (sqrt((1.0d0 + y)) + sqrt((1.0d0 + z)))) - (sqrt(x) + (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 (1.0 + (Math.sqrt((1.0 + y)) + Math.sqrt((1.0 + z)))) - (Math.sqrt(x) + (Math.sqrt(y) + Math.sqrt(z)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (1.0 + (math.sqrt((1.0 + y)) + math.sqrt((1.0 + z)))) - (math.sqrt(x) + (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(1.0 + Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + z)))) - Float64(sqrt(x) + 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 = (1.0 + (sqrt((1.0 + y)) + sqrt((1.0 + z)))) - (sqrt(x) + (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[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in t around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

4. Applied rewrites 33.0%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 33.0

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

7. Applied rewrites 33.0%

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

Alternative 18: 7.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return ((math.sqrt((1.0 + x)) - math.sqrt(x)) + (1.0 - math.sqrt(z))) + (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(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(1.0 - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

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

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lift-sqrt.f64 36.6

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

4. Applied rewrites 36.6%

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

5. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 7.1

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

7. Applied rewrites 7.1%

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

Alternative 19: 2.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lift-sqrt.f64 36.6

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

4. Applied rewrites 36.6%

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

5. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 2.4

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

7. Applied rewrites 2.4%

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

8. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. lift-sqrt.f64 2.4

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

10. Applied rewrites 2.4%

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

Reproduce

herbie shell --seed 2025134 
(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))))