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

Percentage Accurate: 24.3% → 57.0%

Time: 6.0s

Alternatives: 6

Speedup: N/A×

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: 24.3% 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: 57.0% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 2.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(-1 \cdot p\right) \cdot \left(\frac{\left|p\right| + \left(\mathsf{fma}\left(\frac{q\_m \cdot q\_m}{p}, 2, \left|r\right|\right) - r\right)}{p} \cdot -1 - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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 2.6e-23)
   (*
    (/ 1.0 2.0)
    (*
     (* -1.0 p)
     (-
      (* (/ (+ (fabs p) (- (fma (/ (* q_m q_m) p) 2.0 (fabs r)) r)) p) -1.0)
      1.0)))
   (*
    0.5
    (-
     (+ (+ (fabs p) (fabs r)) (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m)))))
     (* 2.0 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 <= 2.6e-23) {
		tmp = (1.0 / 2.0) * ((-1.0 * p) * ((((fabs(p) + (fma(((q_m * q_m) / p), 2.0, fabs(r)) - r)) / p) * -1.0) - 1.0));
	} else {
		tmp = 0.5 * (((fabs(p) + fabs(r)) + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * 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 <= 2.6e-23)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(-1.0 * p) * Float64(Float64(Float64(Float64(abs(p) + Float64(fma(Float64(Float64(q_m * q_m) / p), 2.0, abs(r)) - r)) / p) * -1.0) - 1.0)));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * q_m)));
	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[q$95$m, 2.6e-23], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(-1.0 * p), $MachinePrecision] * N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[(N[(N[(q$95$m * q$95$m), $MachinePrecision] / p), $MachinePrecision] * 2.0 + N[Abs[r], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(p * N[(-0.25 * N[(p / q$95$m), $MachinePrecision] + N[(0.5 * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 2.6e-23

   1. Initial program 22.3%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 22.0%

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

   if 2.6e-23 < q

   1. Initial program 26.5%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 25.7%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 63.5%

      \[\leadsto 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - \color{blue}{2 \cdot q}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 54.8% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.85 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{q\_m \cdot q\_m}{r} - \left(-1 \cdot r\right) \cdot \left(\left(\frac{\left|p\right|}{r} + \left(\frac{\left|r\right|}{r} - 1\right)\right) - -1 \cdot \frac{p}{r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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 1.85e+34)
   (*
    (/ 1.0 2.0)
    (-
     (* -2.0 (/ (* q_m q_m) r))
     (*
      (* -1.0 r)
      (- (+ (/ (fabs p) r) (- (/ (fabs r) r) 1.0)) (* -1.0 (/ p r))))))
   (*
    0.5
    (-
     (+ (+ (fabs p) (fabs r)) (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m)))))
     (* 2.0 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 <= 1.85e+34) {
		tmp = (1.0 / 2.0) * ((-2.0 * ((q_m * q_m) / r)) - ((-1.0 * r) * (((fabs(p) / r) + ((fabs(r) / r) - 1.0)) - (-1.0 * (p / r)))));
	} else {
		tmp = 0.5 * (((fabs(p) + fabs(r)) + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * 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 <= 1.85e+34)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(Float64(-2.0 * Float64(Float64(q_m * q_m) / r)) - Float64(Float64(-1.0 * r) * Float64(Float64(Float64(abs(p) / r) + Float64(Float64(abs(r) / r) - 1.0)) - Float64(-1.0 * Float64(p / r))))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * q_m)));
	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[q$95$m, 1.85e+34], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(-2.0 * N[(N[(q$95$m * q$95$m), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 * r), $MachinePrecision] * N[(N[(N[(N[Abs[p], $MachinePrecision] / r), $MachinePrecision] + N[(N[(N[Abs[r], $MachinePrecision] / r), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(p / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(p * N[(-0.25 * N[(p / q$95$m), $MachinePrecision] + N[(0.5 * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.85000000000000004e34

   1. Initial program 22.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in q around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 21.0%

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

   if 1.85000000000000004e34 < q

   1. Initial program 25.7%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 24.9%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 66.3%

      \[\leadsto 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - \color{blue}{2 \cdot q}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 48.1% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{q\_m \cdot q\_m}{p}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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 3.3e-23)
   (* (/ 1.0 2.0) (* 2.0 (/ (* q_m q_m) p)))
   (*
    0.5
    (-
     (+ (+ (fabs p) (fabs r)) (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m)))))
     (* 2.0 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 <= 3.3e-23) {
		tmp = (1.0 / 2.0) * (2.0 * ((q_m * q_m) / p));
	} else {
		tmp = 0.5 * (((fabs(p) + fabs(r)) + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * 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 <= 3.3e-23)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(2.0 * Float64(Float64(q_m * q_m) / p)));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * q_m)));
	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[q$95$m, 3.3e-23], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(2.0 * N[(N[(q$95$m * q$95$m), $MachinePrecision] / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(p * N[(-0.25 * N[(p / q$95$m), $MachinePrecision] + N[(0.5 * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.30000000000000021e-23

   1. Initial program 22.3%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in p around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 26.2

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

   8. Applied rewrites 26.2%

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

   if 3.30000000000000021e-23 < q

   1. Initial program 26.5%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 25.7%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 63.5%

      \[\leadsto 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - \color{blue}{2 \cdot q}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 39.4% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)\\ t_1 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;\frac{1}{2} \cdot \left(t\_1 - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q\_m}^{2}}\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\left(t\_1 + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left({\left({t\_0}^{-1}\right)}^{0.5}, r \cdot p, \left|r\right| + \left(\left|p\right| - {t\_0}^{0.5}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma (* q_m q_m) 4.0 (* p p))) (t_1 (+ (fabs p) (fabs r))))
   (if (<=
        (*
         (/ 1.0 2.0)
         (- t_1 (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q_m 2.0))))))
        (- INFINITY))
     (*
      0.5
      (- (+ t_1 (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m))))) (* 2.0 q_m)))
     (*
      0.5
      (fma
       (pow (pow t_0 -1.0) 0.5)
       (* r p)
       (+ (fabs r) (- (fabs p) (pow t_0 0.5))))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fma((q_m * q_m), 4.0, (p * p));
	double t_1 = fabs(p) + fabs(r);
	double tmp;
	if (((1.0 / 2.0) * (t_1 - sqrt((pow((p - r), 2.0) + (4.0 * pow(q_m, 2.0)))))) <= -((double) INFINITY)) {
		tmp = 0.5 * ((t_1 + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * q_m));
	} else {
		tmp = 0.5 * fma(pow(pow(t_0, -1.0), 0.5), (r * p), (fabs(r) + (fabs(p) - pow(t_0, 0.5))));
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = fma(Float64(q_m * q_m), 4.0, Float64(p * p))
	t_1 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (Float64(Float64(1.0 / 2.0) * Float64(t_1 - sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q_m ^ 2.0)))))) <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(Float64(t_1 + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * q_m)));
	else
		tmp = Float64(0.5 * fma(((t_0 ^ -1.0) ^ 0.5), Float64(r * p), Float64(abs(r) + Float64(abs(p) - (t_0 ^ 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_] := Block[{t$95$0 = N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$1 - N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(0.5 * N[(N[(t$95$1 + N[(p * N[(-0.25 * N[(p / q$95$m), $MachinePrecision] + N[(0.5 * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[N[Power[t$95$0, -1.0], $MachinePrecision], 0.5], $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] - N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (-.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64))))))) < -inf.0

   1. Initial program 4.0%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 3.6%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - \color{blue}{2 \cdot q}\right) \]

   if -inf.0 < (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (-.f64 (+.f64 (fabs.f64 p) (fabs.f64 r)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 p r) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (pow.f64 q #s(literal 2 binary64)))))))

   1. Initial program 52.1%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      5. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right) \]

      10. associate--l+ N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      12. lift-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      13. lower--.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      14. lift-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      15. lift-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

   7. Applied rewrites 42.3%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 37.9% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{-1}{p}, r \cdot p, \left|r\right| + 2 \cdot q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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 7.2e-66)
   (* 0.5 (fma (/ -1.0 p) (* r p) (+ (fabs r) (* 2.0 q_m))))
   (*
    0.5
    (-
     (+ (+ (fabs p) (fabs r)) (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m)))))
     (* 2.0 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 <= 7.2e-66) {
		tmp = 0.5 * fma((-1.0 / p), (r * p), (fabs(r) + (2.0 * q_m)));
	} else {
		tmp = 0.5 * (((fabs(p) + fabs(r)) + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * 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 <= 7.2e-66)
		tmp = Float64(0.5 * fma(Float64(-1.0 / p), Float64(r * p), Float64(abs(r) + Float64(2.0 * q_m))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * q_m)));
	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[q$95$m, 7.2e-66], N[(0.5 * N[(N[(-1.0 / p), $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(p * N[(-0.25 * N[(p / q$95$m), $MachinePrecision] + N[(0.5 * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 7.20000000000000025e-66

   1. Initial program 22.0%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      5. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right) \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right) \]

      10. associate--l+ N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      12. lift-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      13. lower--.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      14. lift-fabs.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\left(q \cdot q\right) \cdot 4 + p \cdot p\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      15. lift-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right) \]

   7. Applied rewrites 16.4%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left|r\right| + \left(\left|p\right| - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)\right) \]

   8. Taylor expanded in q around -inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{\frac{1}{2}}, r \cdot p, \left|r\right| + 2 \cdot q\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 31.1

         \[\leadsto 0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left|r\right| + 2 \cdot q\right) \]

   10. Applied rewrites 31.1%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left|r\right| + 2 \cdot q\right) \]

   11. Taylor expanded in p around -inf

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

       1. lower-/.f64 31.5

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\frac{-1}{p}, r \cdot p, \left|r\right| + 2 \cdot q\right) \]

   13. Applied rewrites 31.5%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\frac{-1}{p}, \color{blue}{r} \cdot p, \left|r\right| + 2 \cdot q\right) \]

   if 7.20000000000000025e-66 < q

   1. Initial program 26.5%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 25.8%

      \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]

   6. Taylor expanded in p around 0

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 59.6%

      \[\leadsto 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - \color{blue}{2 \cdot q}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.8% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right) \end{array} \]

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
 (*
  0.5
  (-
   (+ (+ (fabs p) (fabs r)) (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m)))))
   (* 2.0 q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return 0.5 * (((fabs(p) + fabs(r)) + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * q_m));
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(0.5 * Float64(Float64(Float64(abs(p) + abs(r)) + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * 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_] := N[(0.5 * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[(p * N[(-0.25 * N[(p / q$95$m), $MachinePrecision] + N[(0.5 * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.4%

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

3. Taylor expanded in r around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-out N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 16.8%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left({\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}\right)}^{0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - {\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{0.5}\right)} \]

6. Taylor expanded in p around 0

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

   1. metadata-eval N/A

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

   2. lower--.f64 N/A

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

8. Applied rewrites 19.4%

   \[\leadsto 0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - \color{blue}{2 \cdot q}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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)))))))