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

Percentage Accurate: 23.5% → 57.8%

Time: 7.2s

Alternatives: 9

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: 23.5% 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: 57.8% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{2} \cdot \left(p \cdot \left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p} - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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 6.4e-12)
   (* (/ 1.0 2.0) (* p (- (/ (+ (fabs p) (- (fabs r) r)) p) -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 <= 6.4e-12) {
		tmp = (1.0 / 2.0) * (p * (((fabs(p) + (fabs(r) - r)) / p) - -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 <= 6.4e-12)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(p * Float64(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p) - -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, 6.4e-12], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(p * N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $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 < 6.4000000000000002e-12

   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 p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-fabs.f64 22.3

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

   5. Applied rewrites 22.3%

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

   if 6.4000000000000002e-12 < q

   1. Initial program 28.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 28.2%

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

   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 67.4%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{2} \cdot \left(p \cdot \left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p} - -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q}, 0.5 \cdot \frac{r}{q}\right)\right) - 2 \cdot q\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.3% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|p\right| + \left|r\right|\right) + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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 7e-13)
   (* (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5) 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 <= 7e-13) {
		tmp = ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5) * 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 <= 7e-13)
		tmp = Float64(Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5) * 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, 7e-13], N[(N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * r), $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 < 7.0000000000000005e-13

   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

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 9.7%

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

   6. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 21.1

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

   8. Applied rewrites 21.1%

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

   if 7.0000000000000005e-13 < q

   1. Initial program 28.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 28.2%

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

   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 67.4%

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

Alternative 3: 45.0% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;q\_m \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \frac{p}{r}, 0.5 \cdot \frac{t\_0}{r}\right) - 0.5\right) \cdot r\\ \mathbf{elif}\;q\_m \leq 1.56 \cdot 10^{-75}:\\ \;\;\;\;\left(q\_m \cdot q\_m\right) \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q\_m \cdot q\_m} - {r}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= q_m 2.15e-162)
     (* (- (fma 0.5 (/ p r) (* 0.5 (/ t_0 r))) 0.5) r)
     (if (<= q_m 1.56e-75)
       (*
        (* q_m q_m)
        (-
         (/
          (* r (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5))
          (* q_m q_m))
         (pow r -1.0)))
       (*
        0.5
        (-
         (+ t_0 (* 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 t_0 = fabs(p) + fabs(r);
	double tmp;
	if (q_m <= 2.15e-162) {
		tmp = (fma(0.5, (p / r), (0.5 * (t_0 / r))) - 0.5) * r;
	} else if (q_m <= 1.56e-75) {
		tmp = (q_m * q_m) * (((r * ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - pow(r, -1.0));
	} else {
		tmp = 0.5 * ((t_0 + (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)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (q_m <= 2.15e-162)
		tmp = Float64(Float64(fma(0.5, Float64(p / r), Float64(0.5 * Float64(t_0 / r))) - 0.5) * r);
	elseif (q_m <= 1.56e-75)
		tmp = Float64(Float64(q_m * q_m) * Float64(Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5)) / Float64(q_m * q_m)) - (r ^ -1.0)));
	else
		tmp = Float64(0.5 * Float64(Float64(t_0 + 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_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 2.15e-162], N[(N[(N[(0.5 * N[(p / r), $MachinePrecision] + N[(0.5 * N[(t$95$0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[q$95$m, 1.56e-75], N[(N[(q$95$m * q$95$m), $MachinePrecision] * N[(N[(N[(r * N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] - N[Power[r, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(t$95$0 + 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 3 regimes

2. if q < 2.14999999999999998e-162

   1. Initial program 25.8%

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

   3. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lift-fabs.f64 21.2

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

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in p around 0

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. metadata-eval N/A

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

       5. lift-/.f64 N/A

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

       6. div-add-rev N/A

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

       7. lower-*.f64 N/A

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

       8. metadata-eval N/A

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

       9. div-add-rev N/A

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

       10. lower-/.f64 N/A

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

       11. lower-+.f64 N/A

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

       12. lift-fabs.f64 N/A

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

       13. lift-fabs.f64 10.5

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

   11. Applied rewrites 10.5%

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

   if 2.14999999999999998e-162 < q < 1.5600000000000001e-75

   1. Initial program 6.1%

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

   3. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 14.9%

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

   if 1.5600000000000001e-75 < q

   1. Initial program 25.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 25.6%

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

   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 58.8%

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

Alternative 4: 39.2% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;\frac{1}{2} \cdot \left(t\_0 - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q\_m}^{2}}\right) \leq -2 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 + p \cdot \mathsf{fma}\left(-0.25, \frac{p}{q\_m}, 0.5 \cdot \frac{r}{q\_m}\right)\right) - 2 \cdot q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(q\_m \cdot q\_m\right) \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q\_m \cdot q\_m} - {r}^{-1}\right)\\ \end{array} \end{array} \]

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 (+ (fabs p) (fabs r))))
   (if (<=
        (*
         (/ 1.0 2.0)
         (- t_0 (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q_m 2.0))))))
        -2e-122)
     (*
      0.5
      (- (+ t_0 (* p (fma -0.25 (/ p q_m) (* 0.5 (/ r q_m))))) (* 2.0 q_m)))
     (*
      (* q_m q_m)
      (-
       (/ (* r (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5)) (* q_m q_m))
       (pow r -1.0))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (((1.0 / 2.0) * (t_0 - sqrt((pow((p - r), 2.0) + (4.0 * pow(q_m, 2.0)))))) <= -2e-122) {
		tmp = 0.5 * ((t_0 + (p * fma(-0.25, (p / q_m), (0.5 * (r / q_m))))) - (2.0 * q_m));
	} else {
		tmp = (q_m * q_m) * (((r * ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - pow(r, -1.0));
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (Float64(Float64(1.0 / 2.0) * Float64(t_0 - sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q_m ^ 2.0)))))) <= -2e-122)
		tmp = Float64(0.5 * Float64(Float64(t_0 + Float64(p * fma(-0.25, Float64(p / q_m), Float64(0.5 * Float64(r / q_m))))) - Float64(2.0 * q_m)));
	else
		tmp = Float64(Float64(q_m * q_m) * Float64(Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5)) / Float64(q_m * q_m)) - (r ^ -1.0)));
	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[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 - 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], -2e-122], N[(0.5 * N[(N[(t$95$0 + 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[(N[(q$95$m * q$95$m), $MachinePrecision] * N[(N[(N[(r * N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] - N[Power[r, -1.0], $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))))))) < -2.00000000000000012e-122

   1. Initial program 23.9%

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

   3. Taylor expanded in r around 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 23.5%

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

   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 28.1%

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

   if -2.00000000000000012e-122 < (*.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 25.6%

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

   3. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 12.0%

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

Alternative 5: 39.7% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q\_m}^{2}}\right) \leq -2 \cdot 10^{-122}:\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;\left(q\_m \cdot q\_m\right) \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q\_m \cdot q\_m} - {r}^{-1}\right)\\ \end{array} \end{array} \]

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 (<=
      (*
       (/ 1.0 2.0)
       (-
        (+ (fabs p) (fabs r))
        (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q_m 2.0))))))
      -2e-122)
   (- q_m)
   (*
    (* q_m q_m)
    (-
     (/ (* r (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5)) (* q_m q_m))
     (pow r -1.0)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (((1.0 / 2.0) * ((fabs(p) + fabs(r)) - sqrt((pow((p - r), 2.0) + (4.0 * pow(q_m, 2.0)))))) <= -2e-122) {
		tmp = -q_m;
	} else {
		tmp = (q_m * q_m) * (((r * ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - pow(r, -1.0));
	}
	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 (((1.0d0 / 2.0d0) * ((abs(p) + abs(r)) - sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q_m ** 2.0d0)))))) <= (-2d-122)) then
        tmp = -q_m
    else
        tmp = (q_m * q_m) * (((r * ((0.5d0 * (((p + abs(p)) + abs(r)) / r)) - 0.5d0)) / (q_m * q_m)) - (r ** (-1.0d0)))
    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 (((1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) - Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q_m, 2.0)))))) <= -2e-122) {
		tmp = -q_m;
	} else {
		tmp = (q_m * q_m) * (((r * ((0.5 * (((p + Math.abs(p)) + Math.abs(r)) / r)) - 0.5)) / (q_m * q_m)) - Math.pow(r, -1.0));
	}
	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 ((1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) - math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q_m, 2.0)))))) <= -2e-122:
		tmp = -q_m
	else:
		tmp = (q_m * q_m) * (((r * ((0.5 * (((p + math.fabs(p)) + math.fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - math.pow(r, -1.0))
	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 (Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) - sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q_m ^ 2.0)))))) <= -2e-122)
		tmp = Float64(-q_m);
	else
		tmp = Float64(Float64(q_m * q_m) * Float64(Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5)) / Float64(q_m * q_m)) - (r ^ -1.0)));
	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 (((1.0 / 2.0) * ((abs(p) + abs(r)) - sqrt((((p - r) ^ 2.0) + (4.0 * (q_m ^ 2.0)))))) <= -2e-122)
		tmp = -q_m;
	else
		tmp = (q_m * q_m) * (((r * ((0.5 * (((p + abs(p)) + abs(r)) / r)) - 0.5)) / (q_m * q_m)) - (r ^ -1.0));
	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[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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-122], (-q$95$m), N[(N[(q$95$m * q$95$m), $MachinePrecision] * N[(N[(N[(r * N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] - N[Power[r, -1.0], $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))))))) < -2.00000000000000012e-122

   1. Initial program 23.9%

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

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{-1 \cdot q} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(q\right) \]

      2. lower-neg.f64 28.5

         \[\leadsto -q \]

   5. Applied rewrites 28.5%

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

   if -2.00000000000000012e-122 < (*.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 25.6%

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

   3. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 12.0%

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

Alternative 6: 18.5% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \left(q\_m \cdot q\_m\right) \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q\_m \cdot q\_m} - {r}^{-1}\right) \end{array} \]

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
 (*
  (* q_m q_m)
  (-
   (/ (* r (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5)) (* q_m q_m))
   (pow r -1.0))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return (q_m * q_m) * (((r * ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - pow(r, -1.0));
}

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 * q_m) * (((r * ((0.5d0 * (((p + abs(p)) + abs(r)) / r)) - 0.5d0)) / (q_m * q_m)) - (r ** (-1.0d0)))
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 * q_m) * (((r * ((0.5 * (((p + Math.abs(p)) + Math.abs(r)) / r)) - 0.5)) / (q_m * q_m)) - Math.pow(r, -1.0));
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return (q_m * q_m) * (((r * ((0.5 * (((p + math.fabs(p)) + math.fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - math.pow(r, -1.0))

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(Float64(q_m * q_m) * Float64(Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5)) / Float64(q_m * q_m)) - (r ^ -1.0)))
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 * q_m) * (((r * ((0.5 * (((p + abs(p)) + abs(r)) / r)) - 0.5)) / (q_m * q_m)) - (r ^ -1.0));
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[(N[(q$95$m * q$95$m), $MachinePrecision] * N[(N[(N[(r * N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] - N[Power[r, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in r around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 7.9%

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

6. Taylor expanded in q around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower--.f64 N/A

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

8. Applied rewrites 8.9%

   \[\leadsto \left(q \cdot q\right) \cdot \color{blue}{\left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right)} \]
9. Add Preprocessing

Alternative 7: 17.6% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ e^{\log q\_m \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q\_m \cdot q\_m} - {r}^{-1}\right) \end{array} \]

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
 (*
  (exp (* (log q_m) 2.0))
  (-
   (/ (* r (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5)) (* q_m q_m))
   (pow r -1.0))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return exp((log(q_m) * 2.0)) * (((r * ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - pow(r, -1.0));
}

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 = exp((log(q_m) * 2.0d0)) * (((r * ((0.5d0 * (((p + abs(p)) + abs(r)) / r)) - 0.5d0)) / (q_m * q_m)) - (r ** (-1.0d0)))
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 Math.exp((Math.log(q_m) * 2.0)) * (((r * ((0.5 * (((p + Math.abs(p)) + Math.abs(r)) / r)) - 0.5)) / (q_m * q_m)) - Math.pow(r, -1.0));
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return math.exp((math.log(q_m) * 2.0)) * (((r * ((0.5 * (((p + math.fabs(p)) + math.fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - math.pow(r, -1.0))

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(exp(Float64(log(q_m) * 2.0)) * Float64(Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5)) / Float64(q_m * q_m)) - (r ^ -1.0)))
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = exp((log(q_m) * 2.0)) * (((r * ((0.5 * (((p + abs(p)) + abs(r)) / r)) - 0.5)) / (q_m * q_m)) - (r ^ -1.0));
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[(N[Exp[N[(N[Log[q$95$m], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(r * N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] - N[Power[r, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in r around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 7.9%

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

6. Taylor expanded in q around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower--.f64 N/A

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

8. Applied rewrites 8.9%

   \[\leadsto \left(q \cdot q\right) \cdot \color{blue}{\left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 4.0

      \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right) \]

10. Applied rewrites 4.0%

    \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {\color{blue}{r}}^{-1}\right) \]
11. Add Preprocessing

Alternative 8: 15.7% accurate, N/A× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := e^{\log q\_m \cdot 2}\\ \mathbf{if}\;r \leq 1.16 \cdot 10^{-250}:\\ \;\;\;\;t\_0 \cdot \frac{r \cdot \left(0.5 \cdot \frac{p + \left(\left|p\right| + \left|r\right|\right)}{r} - 0.5\right)}{q\_m \cdot q\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q\_m \cdot q\_m} - e^{\log r \cdot -1}\right)\\ \end{array} \end{array} \]

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 (exp (* (log q_m) 2.0))))
   (if (<= r 1.16e-250)
     (*
      t_0
      (/ (* r (- (* 0.5 (/ (+ p (+ (fabs p) (fabs r))) r)) 0.5)) (* q_m q_m)))
     (*
      t_0
      (-
       (/ (* r (- (* 0.5 (/ (+ (+ p (fabs p)) (fabs r)) r)) 0.5)) (* q_m q_m))
       (exp (* (log r) -1.0)))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = exp((log(q_m) * 2.0));
	double tmp;
	if (r <= 1.16e-250) {
		tmp = t_0 * ((r * ((0.5 * ((p + (fabs(p) + fabs(r))) / r)) - 0.5)) / (q_m * q_m));
	} else {
		tmp = t_0 * (((r * ((0.5 * (((p + fabs(p)) + fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - exp((log(r) * -1.0)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp((log(q_m) * 2.0d0))
    if (r <= 1.16d-250) then
        tmp = t_0 * ((r * ((0.5d0 * ((p + (abs(p) + abs(r))) / r)) - 0.5d0)) / (q_m * q_m))
    else
        tmp = t_0 * (((r * ((0.5d0 * (((p + abs(p)) + abs(r)) / r)) - 0.5d0)) / (q_m * q_m)) - exp((log(r) * (-1.0d0))))
    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 t_0 = Math.exp((Math.log(q_m) * 2.0));
	double tmp;
	if (r <= 1.16e-250) {
		tmp = t_0 * ((r * ((0.5 * ((p + (Math.abs(p) + Math.abs(r))) / r)) - 0.5)) / (q_m * q_m));
	} else {
		tmp = t_0 * (((r * ((0.5 * (((p + Math.abs(p)) + Math.abs(r)) / r)) - 0.5)) / (q_m * q_m)) - Math.exp((Math.log(r) * -1.0)));
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	t_0 = math.exp((math.log(q_m) * 2.0))
	tmp = 0
	if r <= 1.16e-250:
		tmp = t_0 * ((r * ((0.5 * ((p + (math.fabs(p) + math.fabs(r))) / r)) - 0.5)) / (q_m * q_m))
	else:
		tmp = t_0 * (((r * ((0.5 * (((p + math.fabs(p)) + math.fabs(r)) / r)) - 0.5)) / (q_m * q_m)) - math.exp((math.log(r) * -1.0)))
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = exp(Float64(log(q_m) * 2.0))
	tmp = 0.0
	if (r <= 1.16e-250)
		tmp = Float64(t_0 * Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(p + Float64(abs(p) + abs(r))) / r)) - 0.5)) / Float64(q_m * q_m)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(Float64(p + abs(p)) + abs(r)) / r)) - 0.5)) / Float64(q_m * q_m)) - exp(Float64(log(r) * -1.0))));
	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)
	t_0 = exp((log(q_m) * 2.0));
	tmp = 0.0;
	if (r <= 1.16e-250)
		tmp = t_0 * ((r * ((0.5 * ((p + (abs(p) + abs(r))) / r)) - 0.5)) / (q_m * q_m));
	else
		tmp = t_0 * (((r * ((0.5 * (((p + abs(p)) + abs(r)) / r)) - 0.5)) / (q_m * q_m)) - exp((log(r) * -1.0)));
	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_] := Block[{t$95$0 = N[Exp[N[(N[Log[q$95$m], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[r, 1.16e-250], N[(t$95$0 * N[(N[(r * N[(N[(0.5 * N[(N[(p + N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(r * N[(N[(0.5 * N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] - N[Exp[N[(N[Log[r], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.16e-250

   1. Initial program 30.3%

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

   3. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 4.5%

      \[\leadsto \left(q \cdot q\right) \cdot \color{blue}{\left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 2.5

         \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right) \]

   10. Applied rewrites 2.5%

       \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {\color{blue}{r}}^{-1}\right) \]

   11. Taylor expanded in q around 0

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. lower-/.f64 N/A

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

   13. Applied rewrites 1.1%

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

   if 1.16e-250 < r

   1. Initial program 17.0%

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

   3. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 12.3%

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

   6. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 14.3%

      \[\leadsto \left(q \cdot q\right) \cdot \color{blue}{\left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 5.9

         \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right) \]

   10. Applied rewrites 5.9%

       \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {\color{blue}{r}}^{-1}\right) \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

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

       2. pow-to-exp N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-log.f64 5.7

          \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - e^{\log r \cdot -1}\right) \]

   12. Applied rewrites 5.7%

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

Alternative 9: 5.8% accurate, N/A× speedup

Math

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

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
 (*
  (exp (* (log q_m) 2.0))
  (/ (* r (- (* 0.5 (/ (+ p (+ (fabs p) (fabs r))) r)) 0.5)) (* q_m q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return exp((log(q_m) * 2.0)) * ((r * ((0.5 * ((p + (fabs(p) + fabs(r))) / r)) - 0.5)) / (q_m * 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 = exp((log(q_m) * 2.0d0)) * ((r * ((0.5d0 * ((p + (abs(p) + abs(r))) / r)) - 0.5d0)) / (q_m * 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 Math.exp((Math.log(q_m) * 2.0)) * ((r * ((0.5 * ((p + (Math.abs(p) + Math.abs(r))) / r)) - 0.5)) / (q_m * q_m));
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return math.exp((math.log(q_m) * 2.0)) * ((r * ((0.5 * ((p + (math.fabs(p) + math.fabs(r))) / r)) - 0.5)) / (q_m * q_m))

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(exp(Float64(log(q_m) * 2.0)) * Float64(Float64(r * Float64(Float64(0.5 * Float64(Float64(p + Float64(abs(p) + abs(r))) / r)) - 0.5)) / Float64(q_m * 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 = exp((log(q_m) * 2.0)) * ((r * ((0.5 * ((p + (abs(p) + abs(r))) / r)) - 0.5)) / (q_m * 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[(N[Exp[N[(N[Log[q$95$m], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(r * N[(N[(0.5 * N[(N[(p + N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in r around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 7.9%

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

6. Taylor expanded in q around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower--.f64 N/A

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

8. Applied rewrites 8.9%

   \[\leadsto \left(q \cdot q\right) \cdot \color{blue}{\left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 4.0

      \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {r}^{-1}\right) \]

10. Applied rewrites 4.0%

    \[\leadsto e^{\log q \cdot 2} \cdot \left(\frac{r \cdot \left(0.5 \cdot \frac{\left(p + \left|p\right|\right) + \left|r\right|}{r} - 0.5\right)}{q \cdot q} - {\color{blue}{r}}^{-1}\right) \]

11. Taylor expanded in q around 0

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

    1. metadata-eval N/A

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

    2. metadata-eval N/A

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

    3. lower-/.f64 N/A

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

13. Applied rewrites 1.4%

    \[\leadsto e^{\log q \cdot 2} \cdot \frac{r \cdot \left(0.5 \cdot \frac{p + \left(\left|p\right| + \left|r\right|\right)}{r} - 0.5\right)}{q \cdot \color{blue}{q}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))))