1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))

Percentage Accurate: 45.5% → 81.9%

Time: 7.4s

Alternatives: 9

Speedup: 35.6×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))

C

double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
}

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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
end function

Java

public static double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
}

Python

def code(p, r, q):
	return (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))

Julia

function code(p, r, q)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
end

Wolfram

code[p_, r_, q_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 45.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))

C

double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
}

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(p, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
end function

Java

public static double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
}

Python

def code(p, r, q):
	return (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))

Julia

function code(p, r, q)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
end

Wolfram

code[p_, r_, q_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9 \cdot 10^{+131}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q\_m + q\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 9e+131)
   (* (+ (+ (fabs r) (fabs p)) (- r p)) 0.5)
   (* 0.5 (+ (+ (fabs p) (fabs r)) (+ q_m q_m)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e+131) {
		tmp = ((fabs(r) + fabs(p)) + (r - p)) * 0.5;
	} else {
		tmp = 0.5 * ((fabs(p) + fabs(r)) + (q_m + q_m));
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 9d+131) then
        tmp = ((abs(r) + abs(p)) + (r - p)) * 0.5d0
    else
        tmp = 0.5d0 * ((abs(p) + abs(r)) + (q_m + q_m))
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e+131) {
		tmp = ((Math.abs(r) + Math.abs(p)) + (r - p)) * 0.5;
	} else {
		tmp = 0.5 * ((Math.abs(p) + Math.abs(r)) + (q_m + q_m));
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 9e+131:
		tmp = ((math.fabs(r) + math.fabs(p)) + (r - p)) * 0.5
	else:
		tmp = 0.5 * ((math.fabs(p) + math.fabs(r)) + (q_m + q_m))
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 9e+131)
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) + Float64(r - p)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) + Float64(q_m + q_m)));
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 9e+131)
		tmp = ((abs(r) + abs(p)) + (r - p)) * 0.5;
	else
		tmp = 0.5 * ((abs(p) + abs(r)) + (q_m + q_m));
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 9e+131], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(r - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(q$95$m + q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 9.00000000000000039e131

   1. Initial program 49.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r \cdot \left(1 + -1 \cdot \frac{p}{r}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(1 + -1 \cdot \frac{p}{r}\right) \cdot \color{blue}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(1 + -1 \cdot \frac{p}{r}\right) \cdot \color{blue}{r}\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-1 \cdot \frac{p}{r} + 1\right) \cdot r\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\frac{p}{r} \cdot -1 + 1\right) \cdot r\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r\right) \]

      6. lower-/.f64 30.6

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r\right) \]

   5. Applied rewrites 30.6%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r}\right) \]

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r + \color{blue}{-1 \cdot p}\right)\right) \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{p}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - 1 \cdot p\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot p\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot {p}^{1}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot {p}^{\left(\frac{2}{2}\right)}\right)\right) \]

      6. sqrt-pow1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \sqrt{{p}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \sqrt{p \cdot p}\right)\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \left|p\right|\right)\right) \]

      9. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1 \cdot p\right|\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|\mathsf{neg}\left(p\right)\right|\right)\right) \]

      11. neg-fabs N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|p\right|\right)\right) \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \sqrt{p \cdot p}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \sqrt{{p}^{2}}\right)\right) \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - {p}^{\left(\frac{2}{\color{blue}{2}}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - {p}^{1}\right)\right) \]

      16. unpow1 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

      17. lower--.f64 38.1

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

   8. Applied rewrites 38.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2}} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      6. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left|r\right| + \left|p\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left|r\right| + \left|p\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      10. lift-fabs.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left|r\right|} + \left|p\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      11. lift-fabs.f64 N/A

          \[\leadsto \left(\left(\left|r\right| + \color{blue}{\left|p\right|}\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      12. metadata-eval 38.1

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 38.1%

       \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot 0.5} \]

   if 9.00000000000000039e131 < q

   1. Initial program 13.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{2 \cdot q}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot \color{blue}{2}\right) \]

      2. lower-*.f64 88.2

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot \color{blue}{2}\right) \]

   5. Applied rewrites 88.2%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{q \cdot 2}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]

      2. metadata-eval 88.2

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]

   7. Applied rewrites 88.2%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + 2 \cdot \color{blue}{q}\right) \]

      3. count-2-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q + \color{blue}{q}\right)\right) \]

      4. lower-+.f64 88.2

         \[\leadsto 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q + \color{blue}{q}\right)\right) \]

   9. Applied rewrites 88.2%

      \[\leadsto 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q + \color{blue}{q}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 61.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq -1.06 \cdot 10^{-170} \lor \neg \left(p \leq -1.2 \cdot 10^{-248} \lor \neg \left(p \leq 2.8 \cdot 10^{-207}\right)\right):\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, r\right) + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -1.5e-61)
   (* 0.5 (+ (fabs p) (- p)))
   (if (or (<= p -1.06e-170) (not (or (<= p -1.2e-248) (not (<= p 2.8e-207)))))
     (* (+ (fma q_m 2.0 r) p) 0.5)
     r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -1.5e-61) {
		tmp = 0.5 * (fabs(p) + -p);
	} else if ((p <= -1.06e-170) || !((p <= -1.2e-248) || !(p <= 2.8e-207))) {
		tmp = (fma(q_m, 2.0, r) + p) * 0.5;
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -1.5e-61)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(-p)));
	elseif ((p <= -1.06e-170) || !((p <= -1.2e-248) || !(p <= 2.8e-207)))
		tmp = Float64(Float64(fma(q_m, 2.0, r) + p) * 0.5);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -1.5e-61], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + (-p)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[p, -1.06e-170], N[Not[Or[LessEqual[p, -1.2e-248], N[Not[LessEqual[p, 2.8e-207]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(q$95$m * 2.0 + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], r]]

Derivation

1. Split input into 3 regimes

2. if p < -1.50000000000000006e-61

   1. Initial program 31.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 17.5%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 17.5

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 17.5%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 18.0

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in p around -inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{-1 \cdot p}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \left(\mathsf{neg}\left(p\right)\right)\right) \]

       2. lower-neg.f64 62.4

          \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(-p\right)\right) \]

   12. Applied rewrites 62.4%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{\left(-p\right)}\right) \]

   if -1.50000000000000006e-61 < p < -1.06000000000000004e-170 or -1.20000000000000002e-248 < p < 2.79999999999999993e-207

   1. Initial program 61.6%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

      \[\leadsto \left(\left(r + 2 \cdot q\right) + p\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot q + r\right) + p\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(q \cdot 2 + r\right) + p\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 35.4

         \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   8. Applied rewrites 35.4%

      \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   if -1.06000000000000004e-170 < p < -1.20000000000000002e-248 or 2.79999999999999993e-207 < p

   1. Initial program 46.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 17.2%

      \[\leadsto r \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq -1.06 \cdot 10^{-170} \lor \neg \left(p \leq -1.2 \cdot 10^{-248} \lor \neg \left(p \leq 2.8 \cdot 10^{-207}\right)\right):\\ \;\;\;\;\left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -3.6 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq -1.06 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + q\_m \cdot 2\right)\\ \mathbf{elif}\;p \leq -1.2 \cdot 10^{-248} \lor \neg \left(p \leq 2.8 \cdot 10^{-207}\right):\\ \;\;\;\;r\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, r\right) + p\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -3.6e+107)
   (* 0.5 (+ (fabs p) (- p)))
   (if (<= p -1.06e-170)
     (* 0.5 (+ (fabs p) (* q_m 2.0)))
     (if (or (<= p -1.2e-248) (not (<= p 2.8e-207)))
       r
       (* (+ (fma q_m 2.0 r) p) 0.5)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -3.6e+107) {
		tmp = 0.5 * (fabs(p) + -p);
	} else if (p <= -1.06e-170) {
		tmp = 0.5 * (fabs(p) + (q_m * 2.0));
	} else if ((p <= -1.2e-248) || !(p <= 2.8e-207)) {
		tmp = r;
	} else {
		tmp = (fma(q_m, 2.0, r) + p) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -3.6e+107)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(-p)));
	elseif (p <= -1.06e-170)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(q_m * 2.0)));
	elseif ((p <= -1.2e-248) || !(p <= 2.8e-207))
		tmp = r;
	else
		tmp = Float64(Float64(fma(q_m, 2.0, r) + p) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -3.6e+107], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -1.06e-170], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[p, -1.2e-248], N[Not[LessEqual[p, 2.8e-207]], $MachinePrecision]], r, N[(N[(N[(q$95$m * 2.0 + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if p < -3.5999999999999998e107

   1. Initial program 23.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 18.7%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 18.7

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 19.1

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 19.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in p around -inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{-1 \cdot p}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \left(\mathsf{neg}\left(p\right)\right)\right) \]

       2. lower-neg.f64 80.4

          \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(-p\right)\right) \]

   12. Applied rewrites 80.4%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{\left(-p\right)}\right) \]

   if -3.5999999999999998e107 < p < -1.06000000000000004e-170

   1. Initial program 44.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 20.1%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 20.1

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 20.1%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 20.8

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 20.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in q around inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{2 \cdot q}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + q \cdot \color{blue}{2}\right) \]

       2. lower-*.f64 28.3

          \[\leadsto 0.5 \cdot \left(\left|p\right| + q \cdot \color{blue}{2}\right) \]

   12. Applied rewrites 28.3%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{q \cdot 2}\right) \]

   if -1.06000000000000004e-170 < p < -1.20000000000000002e-248 or 2.79999999999999993e-207 < p

   1. Initial program 46.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 17.2%

      \[\leadsto r \]

   if -1.20000000000000002e-248 < p < 2.79999999999999993e-207

   1. Initial program 72.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

      \[\leadsto \left(\left(r + 2 \cdot q\right) + p\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot q + r\right) + p\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(q \cdot 2 + r\right) + p\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 38.7

         \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   8. Applied rewrites 38.7%

      \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -3.6 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq -1.06 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + q \cdot 2\right)\\ \mathbf{elif}\;p \leq -1.2 \cdot 10^{-248} \lor \neg \left(p \leq 2.8 \cdot 10^{-207}\right):\\ \;\;\;\;r\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -3.6 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) + \left(-p\right)\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -1.06 \cdot 10^{-170}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q\_m + q\_m\right)\right)\\ \mathbf{elif}\;p \leq -1.2 \cdot 10^{-248}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + r\right)\right)\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-207}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, r\right) + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -3.6e+107)
   (* (+ (+ (fabs r) (fabs p)) (- p)) 0.5)
   (if (<= p -1.06e-170)
     (* 0.5 (+ (+ (fabs p) (fabs r)) (+ q_m q_m)))
     (if (<= p -1.2e-248)
       (* 0.5 (+ (fabs p) (+ (fabs r) r)))
       (if (<= p 2.8e-207) (* (+ (fma q_m 2.0 r) p) 0.5) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -3.6e+107) {
		tmp = ((fabs(r) + fabs(p)) + -p) * 0.5;
	} else if (p <= -1.06e-170) {
		tmp = 0.5 * ((fabs(p) + fabs(r)) + (q_m + q_m));
	} else if (p <= -1.2e-248) {
		tmp = 0.5 * (fabs(p) + (fabs(r) + r));
	} else if (p <= 2.8e-207) {
		tmp = (fma(q_m, 2.0, r) + p) * 0.5;
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -3.6e+107)
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) + Float64(-p)) * 0.5);
	elseif (p <= -1.06e-170)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) + Float64(q_m + q_m)));
	elseif (p <= -1.2e-248)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(abs(r) + r)));
	elseif (p <= 2.8e-207)
		tmp = Float64(Float64(fma(q_m, 2.0, r) + p) * 0.5);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -3.6e+107], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + (-p)), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, -1.06e-170], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(q$95$m + q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -1.2e-248], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 2.8e-207], N[(N[(N[(q$95$m * 2.0 + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], r]]]]

Derivation

1. Split input into 5 regimes

2. if p < -3.5999999999999998e107

   1. Initial program 23.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r \cdot \left(1 + -1 \cdot \frac{p}{r}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(1 + -1 \cdot \frac{p}{r}\right) \cdot \color{blue}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(1 + -1 \cdot \frac{p}{r}\right) \cdot \color{blue}{r}\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-1 \cdot \frac{p}{r} + 1\right) \cdot r\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\frac{p}{r} \cdot -1 + 1\right) \cdot r\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r\right) \]

      6. lower-/.f64 51.1

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r\right) \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r}\right) \]

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r + \color{blue}{-1 \cdot p}\right)\right) \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{p}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - 1 \cdot p\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot p\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot {p}^{1}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot {p}^{\left(\frac{2}{2}\right)}\right)\right) \]

      6. sqrt-pow1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \sqrt{{p}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \sqrt{p \cdot p}\right)\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \left|p\right|\right)\right) \]

      9. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1 \cdot p\right|\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|\mathsf{neg}\left(p\right)\right|\right)\right) \]

      11. neg-fabs N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|p\right|\right)\right) \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \sqrt{p \cdot p}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \sqrt{{p}^{2}}\right)\right) \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - {p}^{\left(\frac{2}{\color{blue}{2}}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - {p}^{1}\right)\right) \]

      16. unpow1 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

      17. lower--.f64 83.5

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

   8. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2}} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      6. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left|r\right| + \left|p\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left|r\right| + \left|p\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      10. lift-fabs.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left|r\right|} + \left|p\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      11. lift-fabs.f64 N/A

          \[\leadsto \left(\left(\left|r\right| + \color{blue}{\left|p\right|}\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      12. metadata-eval 83.5

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 83.5%

       \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot 0.5} \]

   11. Taylor expanded in p around -inf

       \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \color{blue}{-1 \cdot p}\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \cdot \frac{1}{2} \]

       2. lower-neg.f64 81.7

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(-p\right)\right) \cdot 0.5 \]

   13. Applied rewrites 81.7%

       \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \color{blue}{\left(-p\right)}\right) \cdot 0.5 \]

   if -3.5999999999999998e107 < p < -1.06000000000000004e-170

   1. Initial program 44.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{2 \cdot q}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot \color{blue}{2}\right) \]

      2. lower-*.f64 33.3

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot \color{blue}{2}\right) \]

   5. Applied rewrites 33.3%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{q \cdot 2}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]

      2. metadata-eval 33.3

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]

   7. Applied rewrites 33.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot 2\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + q \cdot \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + 2 \cdot \color{blue}{q}\right) \]

      3. count-2-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q + \color{blue}{q}\right)\right) \]

      4. lower-+.f64 33.3

         \[\leadsto 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q + \color{blue}{q}\right)\right) \]

   9. Applied rewrites 33.3%

      \[\leadsto 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(q + \color{blue}{q}\right)\right) \]

   if -1.06000000000000004e-170 < p < -1.20000000000000002e-248

   1. Initial program 54.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 68.7%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 68.7

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 68.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 68.7

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 68.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   if -1.20000000000000002e-248 < p < 2.79999999999999993e-207

   1. Initial program 72.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

      \[\leadsto \left(\left(r + 2 \cdot q\right) + p\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot q + r\right) + p\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(q \cdot 2 + r\right) + p\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 38.7

         \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   8. Applied rewrites 38.7%

      \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   if 2.79999999999999993e-207 < p

   1. Initial program 46.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto r \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) + \left(-p\right)\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + q\_m \cdot 2\right)\\ \mathbf{elif}\;p \leq -1.2 \cdot 10^{-248}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + r\right)\right)\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-207}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, r\right) + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -1e+101)
   (* (+ (+ (fabs r) (fabs p)) (- p)) 0.5)
   (if (<= p -5.2e-140)
     (* 0.5 (+ (fabs p) (* q_m 2.0)))
     (if (<= p -1.2e-248)
       (* 0.5 (+ (fabs p) (+ (fabs r) r)))
       (if (<= p 2.8e-207) (* (+ (fma q_m 2.0 r) p) 0.5) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -1e+101) {
		tmp = ((fabs(r) + fabs(p)) + -p) * 0.5;
	} else if (p <= -5.2e-140) {
		tmp = 0.5 * (fabs(p) + (q_m * 2.0));
	} else if (p <= -1.2e-248) {
		tmp = 0.5 * (fabs(p) + (fabs(r) + r));
	} else if (p <= 2.8e-207) {
		tmp = (fma(q_m, 2.0, r) + p) * 0.5;
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -1e+101)
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) + Float64(-p)) * 0.5);
	elseif (p <= -5.2e-140)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(q_m * 2.0)));
	elseif (p <= -1.2e-248)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(abs(r) + r)));
	elseif (p <= 2.8e-207)
		tmp = Float64(Float64(fma(q_m, 2.0, r) + p) * 0.5);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -1e+101], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + (-p)), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, -5.2e-140], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -1.2e-248], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 2.8e-207], N[(N[(N[(q$95$m * 2.0 + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], r]]]]

Derivation

1. Split input into 5 regimes

2. if p < -9.9999999999999998e100

   1. Initial program 22.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r \cdot \left(1 + -1 \cdot \frac{p}{r}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(1 + -1 \cdot \frac{p}{r}\right) \cdot \color{blue}{r}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(1 + -1 \cdot \frac{p}{r}\right) \cdot \color{blue}{r}\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-1 \cdot \frac{p}{r} + 1\right) \cdot r\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\frac{p}{r} \cdot -1 + 1\right) \cdot r\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r\right) \]

      6. lower-/.f64 50.3

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r\right) \]

   5. Applied rewrites 50.3%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{\mathsf{fma}\left(\frac{p}{r}, -1, 1\right) \cdot r}\right) \]

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r + \color{blue}{-1 \cdot p}\right)\right) \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{p}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - 1 \cdot p\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot p\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot {p}^{1}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot {p}^{\left(\frac{2}{2}\right)}\right)\right) \]

      6. sqrt-pow1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \sqrt{{p}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \sqrt{p \cdot p}\right)\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1\right| \cdot \left|p\right|\right)\right) \]

      9. fabs-mul N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|-1 \cdot p\right|\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|\mathsf{neg}\left(p\right)\right|\right)\right) \]

      11. neg-fabs N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \left|p\right|\right)\right) \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \sqrt{p \cdot p}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - \sqrt{{p}^{2}}\right)\right) \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - {p}^{\left(\frac{2}{\color{blue}{2}}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - {p}^{1}\right)\right) \]

      16. unpow1 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

      17. lower--.f64 82.1

          \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

   8. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2}} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      6. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left|r\right| + \left|p\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left|r\right| + \left|p\right|\right)} + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      10. lift-fabs.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left|r\right|} + \left|p\right|\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      11. lift-fabs.f64 N/A

          \[\leadsto \left(\left(\left|r\right| + \color{blue}{\left|p\right|}\right) + \left(r - p\right)\right) \cdot \frac{1}{2} \]

      12. metadata-eval 82.1

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 82.1%

       \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) + \left(r - p\right)\right) \cdot 0.5} \]

   11. Taylor expanded in p around -inf

       \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \color{blue}{-1 \cdot p}\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \cdot \frac{1}{2} \]

       2. lower-neg.f64 80.3

          \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \left(-p\right)\right) \cdot 0.5 \]

   13. Applied rewrites 80.3%

       \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) + \color{blue}{\left(-p\right)}\right) \cdot 0.5 \]

   if -9.9999999999999998e100 < p < -5.1999999999999996e-140

   1. Initial program 48.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 19.0%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 19.0

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 19.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 19.7

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 19.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in q around inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{2 \cdot q}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + q \cdot \color{blue}{2}\right) \]

       2. lower-*.f64 30.5

          \[\leadsto 0.5 \cdot \left(\left|p\right| + q \cdot \color{blue}{2}\right) \]

   12. Applied rewrites 30.5%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{q \cdot 2}\right) \]

   if -5.1999999999999996e-140 < p < -1.20000000000000002e-248

   1. Initial program 45.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 63.0%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 63.0

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 63.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 63.1

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 63.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   if -1.20000000000000002e-248 < p < 2.79999999999999993e-207

   1. Initial program 72.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

      \[\leadsto \left(\left(r + 2 \cdot q\right) + p\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot q + r\right) + p\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(q \cdot 2 + r\right) + p\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 38.7

         \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   8. Applied rewrites 38.7%

      \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   if 2.79999999999999993e-207 < p

   1. Initial program 46.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto r \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -3.6 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + q\_m \cdot 2\right)\\ \mathbf{elif}\;p \leq -1.2 \cdot 10^{-248}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(\left|r\right| + r\right)\right)\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-207}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, r\right) + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -3.6e+107)
   (* 0.5 (+ (fabs p) (- p)))
   (if (<= p -5.2e-140)
     (* 0.5 (+ (fabs p) (* q_m 2.0)))
     (if (<= p -1.2e-248)
       (* 0.5 (+ (fabs p) (+ (fabs r) r)))
       (if (<= p 2.8e-207) (* (+ (fma q_m 2.0 r) p) 0.5) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -3.6e+107) {
		tmp = 0.5 * (fabs(p) + -p);
	} else if (p <= -5.2e-140) {
		tmp = 0.5 * (fabs(p) + (q_m * 2.0));
	} else if (p <= -1.2e-248) {
		tmp = 0.5 * (fabs(p) + (fabs(r) + r));
	} else if (p <= 2.8e-207) {
		tmp = (fma(q_m, 2.0, r) + p) * 0.5;
	} else {
		tmp = r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -3.6e+107)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(-p)));
	elseif (p <= -5.2e-140)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(q_m * 2.0)));
	elseif (p <= -1.2e-248)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(abs(r) + r)));
	elseif (p <= 2.8e-207)
		tmp = Float64(Float64(fma(q_m, 2.0, r) + p) * 0.5);
	else
		tmp = r;
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -3.6e+107], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -5.2e-140], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -1.2e-248], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 2.8e-207], N[(N[(N[(q$95$m * 2.0 + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], r]]]]

Derivation

1. Split input into 5 regimes

2. if p < -3.5999999999999998e107

   1. Initial program 23.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 18.7%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 18.7

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 19.1

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 19.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in p around -inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{-1 \cdot p}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \left(\mathsf{neg}\left(p\right)\right)\right) \]

       2. lower-neg.f64 80.4

          \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(-p\right)\right) \]

   12. Applied rewrites 80.4%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{\left(-p\right)}\right) \]

   if -3.5999999999999998e107 < p < -5.1999999999999996e-140

   1. Initial program 47.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 18.7%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 18.7

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 19.4

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 19.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in q around inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{2 \cdot q}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + q \cdot \color{blue}{2}\right) \]

       2. lower-*.f64 29.8

          \[\leadsto 0.5 \cdot \left(\left|p\right| + q \cdot \color{blue}{2}\right) \]

   12. Applied rewrites 29.8%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{q \cdot 2}\right) \]

   if -5.1999999999999996e-140 < p < -1.20000000000000002e-248

   1. Initial program 45.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 63.0%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 63.0

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 63.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 63.1

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 63.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   if -1.20000000000000002e-248 < p < 2.79999999999999993e-207

   1. Initial program 72.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

      \[\leadsto \left(\left(r + 2 \cdot q\right) + p\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot q + r\right) + p\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(q \cdot 2 + r\right) + p\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 38.7

         \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   8. Applied rewrites 38.7%

      \[\leadsto \left(\mathsf{fma}\left(q, 2, r\right) + p\right) \cdot 0.5 \]

   if 2.79999999999999993e-207 < p

   1. Initial program 46.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto r \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(\left|p\right| + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq -5.2 \cdot 10^{-140}:\\ \;\;\;\;q\_m\\ \mathbf{elif}\;p \leq -1.2 \cdot 10^{-248}:\\ \;\;\;\;r\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-210}:\\ \;\;\;\;q\_m\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -1.5e-61)
   (* 0.5 (+ (fabs p) (- p)))
   (if (<= p -5.2e-140)
     q_m
     (if (<= p -1.2e-248) r (if (<= p 3.3e-210) q_m r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -1.5e-61) {
		tmp = 0.5 * (fabs(p) + -p);
	} else if (p <= -5.2e-140) {
		tmp = q_m;
	} else if (p <= -1.2e-248) {
		tmp = r;
	} else if (p <= 3.3e-210) {
		tmp = q_m;
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (p <= (-1.5d-61)) then
        tmp = 0.5d0 * (abs(p) + -p)
    else if (p <= (-5.2d-140)) then
        tmp = q_m
    else if (p <= (-1.2d-248)) then
        tmp = r
    else if (p <= 3.3d-210) then
        tmp = q_m
    else
        tmp = r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -1.5e-61) {
		tmp = 0.5 * (Math.abs(p) + -p);
	} else if (p <= -5.2e-140) {
		tmp = q_m;
	} else if (p <= -1.2e-248) {
		tmp = r;
	} else if (p <= 3.3e-210) {
		tmp = q_m;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if p <= -1.5e-61:
		tmp = 0.5 * (math.fabs(p) + -p)
	elif p <= -5.2e-140:
		tmp = q_m
	elif p <= -1.2e-248:
		tmp = r
	elif p <= 3.3e-210:
		tmp = q_m
	else:
		tmp = r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -1.5e-61)
		tmp = Float64(0.5 * Float64(abs(p) + Float64(-p)));
	elseif (p <= -5.2e-140)
		tmp = q_m;
	elseif (p <= -1.2e-248)
		tmp = r;
	elseif (p <= 3.3e-210)
		tmp = q_m;
	else
		tmp = r;
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (p <= -1.5e-61)
		tmp = 0.5 * (abs(p) + -p);
	elseif (p <= -5.2e-140)
		tmp = q_m;
	elseif (p <= -1.2e-248)
		tmp = r;
	elseif (p <= 3.3e-210)
		tmp = q_m;
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -1.5e-61], N[(0.5 * N[(N[Abs[p], $MachinePrecision] + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -5.2e-140], q$95$m, If[LessEqual[p, -1.2e-248], r, If[LessEqual[p, 3.3e-210], q$95$m, r]]]]

Derivation

1. Split input into 3 regimes

2. if p < -1.50000000000000006e-61

   1. Initial program 31.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 17.5%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \color{blue}{r}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

      2. metadata-eval 17.5

         \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]

   7. Applied rewrites 17.5%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + r\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\left|p\right| + \left|r\right|\right) + r\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\left|p\right| + \left|r\right|\right)} + r\right) \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\color{blue}{\left|p\right|} + \left|r\right|\right) + r\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \color{blue}{\left|r\right|}\right) + r\right) \]

      5. associate-+l+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + r\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{\left(\left|r\right| + r\right)}\right) \]

      9. lift-fabs.f64 18.0

         \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(\color{blue}{\left|r\right|} + r\right)\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|p\right| + \left(\left|r\right| + r\right)\right)} \]

   10. Taylor expanded in p around -inf

       \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \color{blue}{-1 \cdot p}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left|p\right| + \left(\mathsf{neg}\left(p\right)\right)\right) \]

       2. lower-neg.f64 62.4

          \[\leadsto 0.5 \cdot \left(\left|p\right| + \left(-p\right)\right) \]

   12. Applied rewrites 62.4%

       \[\leadsto 0.5 \cdot \left(\left|p\right| + \color{blue}{\left(-p\right)}\right) \]

   if -1.50000000000000006e-61 < p < -5.1999999999999996e-140 or -1.20000000000000002e-248 < p < 3.3e-210

   1. Initial program 64.7%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.0%

      \[\leadsto \color{blue}{q} \]

   if -5.1999999999999996e-140 < p < -1.20000000000000002e-248 or 3.3e-210 < p

   1. Initial program 46.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 17.7%

      \[\leadsto r \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.9% accurate, 35.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;q\_m\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (if (<= r 4.8e-33) q_m r))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 4.8e-33) {
		tmp = q_m;
	} else {
		tmp = r;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 4.8d-33) then
        tmp = q_m
    else
        tmp = r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 4.8e-33) {
		tmp = q_m;
	} else {
		tmp = r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 4.8e-33:
		tmp = q_m
	else:
		tmp = r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 4.8e-33)
		tmp = q_m;
	else
		tmp = r;
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 4.8e-33)
		tmp = q_m;
	else
		tmp = r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 4.8e-33], q$95$m, r]

Derivation

1. Split input into 2 regimes

2. if r < 4.8e-33

   1. Initial program 49.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q} \]
   4. Step-by-step derivation

   5. Applied rewrites 20.6%

      \[\leadsto \color{blue}{q} \]

   if 4.8e-33 < r

   1. Initial program 33.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left|p\right|} + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left|p\right| + \left(\left|r\right| + \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   5. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + r\right) + p\right) \cdot 0.5} \]

   6. Taylor expanded in r around inf

      \[\leadsto r \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

      \[\leadsto r \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 35.1% accurate, 250.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ q\_m \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 q_m)

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return q_m;
}

Fortran

q_m =     private
NOTE: p, r, and q_m 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(p, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = q_m
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return q_m;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return q_m

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return q_m
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = q_m;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := q$95$m

Derivation

1. Initial program 45.0%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
2. Add Preprocessing

3. Taylor expanded in q around inf

   \[\leadsto \color{blue}{q} \]
4. Step-by-step derivation

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{q} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025047 
(FPCore (p r q)
  :name "1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))"
  :precision binary64
  (* (/ 1.0 2.0) (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))