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

Percentage Accurate: 24.3% → 56.8%

Time: 9.1s

Alternatives: 11

Speedup: 83.3×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(p, r, q)
    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: 56.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(2 \cdot q\_m, p\right)\\ \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47} \lor \neg \left(q\_m \leq 2.6 \cdot 10^{+90}\right):\\ \;\;\;\;\mathsf{fma}\left({\left({t\_0}^{-0.5}\right)}^{2}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (hypot (* 2.0 q_m) p)))
   (if (<= q_m 9e-56)
     (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
     (if (or (<= q_m 3.1e+47) (not (<= q_m 2.6e+90)))
       (*
        (fma (pow (pow t_0 -0.5) 2.0) (* r p) (- (+ (fabs r) (fabs p)) t_0))
        0.5)
       (/ (* (- q_m) q_m) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = hypot((2.0 * q_m), p);
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if ((q_m <= 3.1e+47) || !(q_m <= 2.6e+90)) {
		tmp = fma(pow(pow(t_0, -0.5), 2.0), (r * p), ((fabs(r) + fabs(p)) - t_0)) * 0.5;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = hypot(Float64(2.0 * q_m), p)
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif ((q_m <= 3.1e+47) || !(q_m <= 2.6e+90))
		tmp = Float64(fma(((t_0 ^ -0.5) ^ 2.0), Float64(r * p), Float64(Float64(abs(r) + abs(p)) - t_0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * q$95$m), $MachinePrecision] ^ 2 + p ^ 2], $MachinePrecision]}, If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[q$95$m, 3.1e+47], N[Not[LessEqual[q$95$m, 2.6e+90]], $MachinePrecision]], N[(N[(N[Power[N[Power[t$95$0, -0.5], $MachinePrecision], 2.0], $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47 or 2.5999999999999998e90 < q

   1. Initial program 25.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 23.4%

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

   7. Applied rewrites 23.4%

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

   9. Applied rewrites 56.7%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-0.5}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(2 \cdot q, p\right)\right) \cdot 0.5 \]
   10. Step-by-step derivation

   11. Applied rewrites 57.2%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q \leq 3.1 \cdot 10^{+47} \lor \neg \left(q \leq 2.6 \cdot 10^{+90}\right):\\ \;\;\;\;\mathsf{fma}\left({\left({\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{-0.5}\right)}^{2}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(2 \cdot q, p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(2 \cdot q\_m, p\right)\\ \mathbf{if}\;q\_m \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)\right)}^{-0.5}, r \cdot p, t\_0\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{q\_m}, r \cdot p, t\_0\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (- (+ (fabs r) (fabs p)) (hypot (* 2.0 q_m) p))))
   (if (<= q_m 7.8e-56)
     (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
     (if (<= q_m 3.1e+47)
       (* (fma (pow (fma (* q_m q_m) 4.0 (* p p)) -0.5) (* r p) t_0) 0.5)
       (if (<= q_m 2.6e+90)
         (/ (* (- q_m) q_m) r)
         (* (fma (/ -0.5 q_m) (* r p) t_0) 0.5))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = (fabs(r) + fabs(p)) - hypot((2.0 * q_m), p);
	double tmp;
	if (q_m <= 7.8e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = fma(pow(fma((q_m * q_m), 4.0, (p * p)), -0.5), (r * p), t_0) * 0.5;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = fma((-0.5 / q_m), (r * p), t_0) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(Float64(abs(r) + abs(p)) - hypot(Float64(2.0 * q_m), p))
	tmp = 0.0
	if (q_m <= 7.8e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(fma((fma(Float64(q_m * q_m), 4.0, Float64(p * p)) ^ -0.5), Float64(r * p), t_0) * 0.5);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(fma(Float64(-0.5 / q_m), Float64(r * p), t_0) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(2.0 * q$95$m), $MachinePrecision] ^ 2 + p ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 7.8e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], N[(N[(N[Power[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(r * p), $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(-0.5 / q$95$m), $MachinePrecision] * N[(r * p), $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if q < 7.8e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 7.8e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 34.3%

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

   7. Applied rewrites 34.3%

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

   9. Applied rewrites 34.5%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.0%

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

   9. Applied rewrites 67.5%

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

   10. Taylor expanded in q around -inf

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

   12. Applied rewrites 67.5%

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

Alternative 3: 56.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)\\ t_1 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{t\_0}^{-1}}, r \cdot p, t\_1 - \sqrt{t\_0}\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{q\_m}, r \cdot p, t\_1 - \mathsf{hypot}\left(2 \cdot q\_m, p\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (fma (* q_m q_m) 4.0 (* p p))) (t_1 (+ (fabs r) (fabs p))))
   (if (<= q_m 9e-56)
     (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
     (if (<= q_m 3.1e+47)
       (* (fma (sqrt (pow t_0 -1.0)) (* r p) (- t_1 (sqrt t_0))) 0.5)
       (if (<= q_m 2.6e+90)
         (/ (* (- q_m) q_m) r)
         (* (fma (/ -0.5 q_m) (* r p) (- t_1 (hypot (* 2.0 q_m) p))) 0.5))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fma((q_m * q_m), 4.0, (p * p));
	double t_1 = fabs(r) + fabs(p);
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = fma(sqrt(pow(t_0, -1.0)), (r * p), (t_1 - sqrt(t_0))) * 0.5;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = fma((-0.5 / q_m), (r * p), (t_1 - hypot((2.0 * q_m), p))) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = fma(Float64(q_m * q_m), 4.0, Float64(p * p))
	t_1 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(fma(sqrt((t_0 ^ -1.0)), Float64(r * p), Float64(t_1 - sqrt(t_0))) * 0.5);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(fma(Float64(-0.5 / q_m), Float64(r * p), Float64(t_1 - hypot(Float64(2.0 * q_m), p))) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], N[(N[(N[Sqrt[N[Power[t$95$0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(-0.5 / q$95$m), $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(t$95$1 - N[Sqrt[N[(2.0 * q$95$m), $MachinePrecision] ^ 2 + p ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 34.3%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.0%

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

   9. Applied rewrites 67.5%

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

   10. Taylor expanded in q around -inf

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

   12. Applied rewrites 67.5%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \cdot 0.5\\ \mathbf{elif}\;q \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{q}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(2 \cdot q, p\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)\\ t_1 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{t\_0}^{-1}}, r \cdot p, t\_1 - \sqrt{t\_0}\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{q\_m} \cdot \frac{r}{q\_m}, -0.125, \frac{0.5 \cdot t\_1}{q\_m} - 1\right) \cdot q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (fma (* q_m q_m) 4.0 (* p p))) (t_1 (+ (fabs r) (fabs p))))
   (if (<= q_m 9e-56)
     (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
     (if (<= q_m 3.1e+47)
       (* (fma (sqrt (pow t_0 -1.0)) (* r p) (- t_1 (sqrt t_0))) 0.5)
       (if (<= q_m 2.6e+90)
         (/ (* (- q_m) q_m) r)
         (*
          (fma (* (/ r q_m) (/ r q_m)) -0.125 (- (/ (* 0.5 t_1) q_m) 1.0))
          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 = fma((q_m * q_m), 4.0, (p * p));
	double t_1 = fabs(r) + fabs(p);
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = fma(sqrt(pow(t_0, -1.0)), (r * p), (t_1 - sqrt(t_0))) * 0.5;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = fma(((r / q_m) * (r / q_m)), -0.125, (((0.5 * t_1) / q_m) - 1.0)) * 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 = fma(Float64(q_m * q_m), 4.0, Float64(p * p))
	t_1 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(fma(sqrt((t_0 ^ -1.0)), Float64(r * p), Float64(t_1 - sqrt(t_0))) * 0.5);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(fma(Float64(Float64(r / q_m) * Float64(r / q_m)), -0.125, Float64(Float64(Float64(0.5 * t_1) / q_m) - 1.0)) * 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[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], N[(N[(N[Sqrt[N[Power[t$95$0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(N[(r / q$95$m), $MachinePrecision] * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[(N[(0.5 * t$95$1), $MachinePrecision] / q$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 34.3%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in q around inf

      \[\leadsto q \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{r}^{2}}{{q}^{2}} + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.0%

      \[\leadsto \mathsf{fma}\left(\frac{r}{q} \cdot \frac{r}{q}, -0.125, \frac{0.5 \cdot \left(\left|r\right| + \left|p\right|\right)}{q} - 1\right) \cdot \color{blue}{q} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right)}^{-1}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \cdot 0.5\\ \mathbf{elif}\;q \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{q} \cdot \frac{r}{q}, -0.125, \frac{0.5 \cdot \left(\left|r\right| + \left|p\right|\right)}{q} - 1\right) \cdot q\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)\\ t_1 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left({t\_0}^{-0.5}, r \cdot p, t\_1 - \sqrt{t\_0}\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{q\_m}, r \cdot p, t\_1 - \mathsf{hypot}\left(2 \cdot q\_m, p\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (fma (* q_m q_m) 4.0 (* p p))) (t_1 (+ (fabs r) (fabs p))))
   (if (<= q_m 9e-56)
     (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
     (if (<= q_m 3.1e+47)
       (* (fma (pow t_0 -0.5) (* r p) (- t_1 (sqrt t_0))) 0.5)
       (if (<= q_m 2.6e+90)
         (/ (* (- q_m) q_m) r)
         (* (fma (/ -0.5 q_m) (* r p) (- t_1 (hypot (* 2.0 q_m) p))) 0.5))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fma((q_m * q_m), 4.0, (p * p));
	double t_1 = fabs(r) + fabs(p);
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = fma(pow(t_0, -0.5), (r * p), (t_1 - sqrt(t_0))) * 0.5;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = fma((-0.5 / q_m), (r * p), (t_1 - hypot((2.0 * q_m), p))) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = fma(Float64(q_m * q_m), 4.0, Float64(p * p))
	t_1 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(fma((t_0 ^ -0.5), Float64(r * p), Float64(t_1 - sqrt(t_0))) * 0.5);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(fma(Float64(-0.5 / q_m), Float64(r * p), Float64(t_1 - hypot(Float64(2.0 * q_m), p))) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], N[(N[(N[Power[t$95$0, -0.5], $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(-0.5 / q$95$m), $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(t$95$1 - N[Sqrt[N[(2.0 * q$95$m), $MachinePrecision] ^ 2 + p ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 34.3%

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

   7. Applied rewrites 34.3%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.0%

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

   7. Applied rewrites 18.0%

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

   9. Applied rewrites 67.5%

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

   10. Taylor expanded in q around -inf

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

   12. Applied rewrites 67.5%

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

Alternative 6: 57.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{q\_m}, r \cdot p, t\_0 - \sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)}\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{q\_m} \cdot \frac{r}{q\_m}, -0.125, \frac{0.5 \cdot t\_0}{q\_m} - 1\right) \cdot q\_m\\ \end{array} \end{array} \]

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 r) (fabs p))))
   (if (<= q_m 9e-56)
     (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
     (if (<= q_m 3.1e+47)
       (*
        (fma (/ 0.5 q_m) (* r p) (- t_0 (sqrt (fma (* q_m q_m) 4.0 (* p p)))))
        0.5)
       (if (<= q_m 2.6e+90)
         (/ (* (- q_m) q_m) r)
         (*
          (fma (* (/ r q_m) (/ r q_m)) -0.125 (- (/ (* 0.5 t_0) q_m) 1.0))
          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(r) + fabs(p);
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = fma((0.5 / q_m), (r * p), (t_0 - sqrt(fma((q_m * q_m), 4.0, (p * p))))) * 0.5;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = fma(((r / q_m) * (r / q_m)), -0.125, (((0.5 * t_0) / q_m) - 1.0)) * 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(r) + abs(p))
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(fma(Float64(0.5 / q_m), Float64(r * p), Float64(t_0 - sqrt(fma(Float64(q_m * q_m), 4.0, Float64(p * p))))) * 0.5);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(fma(Float64(Float64(r / q_m) * Float64(r / q_m)), -0.125, Float64(Float64(Float64(0.5 * t_0) / q_m) - 1.0)) * 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[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], N[(N[(N[(0.5 / q$95$m), $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(t$95$0 - N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(N[(r / q$95$m), $MachinePrecision] * N[(r / q$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[(N[(0.5 * t$95$0), $MachinePrecision] / q$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 33.8%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in q around inf

      \[\leadsto q \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{r}^{2}}{{q}^{2}} + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.0%

      \[\leadsto \mathsf{fma}\left(\frac{r}{q} \cdot \frac{r}{q}, -0.125, \frac{0.5 \cdot \left(\left|r\right| + \left|p\right|\right)}{q} - 1\right) \cdot \color{blue}{q} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{q\_m}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)}\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(r, 0.5, p \cdot -0.25\right), \frac{p}{q\_m}, r\right) + \left|p\right|\right) - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 9e-56)
   (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
   (if (<= q_m 3.1e+47)
     (*
      (fma
       (/ 0.5 q_m)
       (* r p)
       (- (+ (fabs r) (fabs p)) (sqrt (fma (* q_m q_m) 4.0 (* p p)))))
      0.5)
     (if (<= q_m 2.6e+90)
       (/ (* (- q_m) q_m) r)
       (*
        (- (+ (fma (fma r 0.5 (* p -0.25)) (/ p q_m) r) (fabs p)) (* q_m 2.0))
        0.5)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = fma((0.5 / q_m), (r * p), ((fabs(r) + fabs(p)) - sqrt(fma((q_m * q_m), 4.0, (p * p))))) * 0.5;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = ((fma(fma(r, 0.5, (p * -0.25)), (p / q_m), r) + fabs(p)) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(fma(Float64(0.5 / q_m), Float64(r * p), Float64(Float64(abs(r) + abs(p)) - sqrt(fma(Float64(q_m * q_m), 4.0, Float64(p * p))))) * 0.5);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(Float64(Float64(fma(fma(r, 0.5, Float64(p * -0.25)), Float64(p / q_m), r) + abs(p)) - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], N[(N[(N[(0.5 / q$95$m), $MachinePrecision] * N[(r * p), $MachinePrecision] + N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(N[(N[(r * 0.5 + N[(p * -0.25), $MachinePrecision]), $MachinePrecision] * N[(p / q$95$m), $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 33.8%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 66.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, r, -0.25 \cdot p\right)}{q}, p, \left|r\right|\right) + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 66.3%

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

Alternative 8: 57.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;-q\_m\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(r, 0.5, p \cdot -0.25\right), \frac{p}{q\_m}, r\right) + \left|p\right|\right) - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 9e-56)
   (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
   (if (<= q_m 3.1e+47)
     (- q_m)
     (if (<= q_m 2.6e+90)
       (/ (* (- q_m) q_m) r)
       (*
        (- (+ (fma (fma r 0.5 (* p -0.25)) (/ p q_m) r) (fabs p)) (* q_m 2.0))
        0.5)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if (q_m <= 3.1e+47) {
		tmp = -q_m;
	} else if (q_m <= 2.6e+90) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = ((fma(fma(r, 0.5, (p * -0.25)), (p / q_m), r) + fabs(p)) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif (q_m <= 3.1e+47)
		tmp = Float64(-q_m);
	elseif (q_m <= 2.6e+90)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(Float64(Float64(fma(fma(r, 0.5, Float64(p * -0.25)), Float64(p / q_m), r) + abs(p)) - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 3.1e+47], (-q$95$m), If[LessEqual[q$95$m, 2.6e+90], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(N[(N[(r * 0.5 + N[(p * -0.25), $MachinePrecision]), $MachinePrecision] * N[(p / q$95$m), $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47

   1. Initial program 34.4%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.6

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

   5. Applied rewrites 33.6%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]

   if 2.5999999999999998e90 < q

   1. Initial program 21.7%

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

   3. Taylor expanded in r around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 66.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, r, -0.25 \cdot p\right)}{q}, p, \left|r\right|\right) + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 66.3%

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

Alternative 9: 57.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q\_m \leq 3.1 \cdot 10^{+47} \lor \neg \left(q\_m \leq 2.6 \cdot 10^{+90}\right):\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 9e-56)
   (* 0.5 (+ p (+ (fabs p) (- (fabs r) r))))
   (if (or (<= q_m 3.1e+47) (not (<= q_m 2.6e+90)))
     (- q_m)
     (/ (* (- q_m) q_m) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else if ((q_m <= 3.1e+47) || !(q_m <= 2.6e+90)) {
		tmp = -q_m;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 9d-56) then
        tmp = 0.5d0 * (p + (abs(p) + (abs(r) - r)))
    else if ((q_m <= 3.1d+47) .or. (.not. (q_m <= 2.6d+90))) then
        tmp = -q_m
    else
        tmp = (-q_m * q_m) / r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (Math.abs(p) + (Math.abs(r) - r)));
	} else if ((q_m <= 3.1e+47) || !(q_m <= 2.6e+90)) {
		tmp = -q_m;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 9e-56:
		tmp = 0.5 * (p + (math.fabs(p) + (math.fabs(r) - r)))
	elif (q_m <= 3.1e+47) or not (q_m <= 2.6e+90):
		tmp = -q_m
	else:
		tmp = (-q_m * q_m) / r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	elseif ((q_m <= 3.1e+47) || !(q_m <= 2.6e+90))
		tmp = Float64(-q_m);
	else
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 9e-56)
		tmp = 0.5 * (p + (abs(p) + (abs(r) - r)));
	elseif ((q_m <= 3.1e+47) || ~((q_m <= 2.6e+90)))
		tmp = -q_m;
	else
		tmp = (-q_m * q_m) / r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[q$95$m, 3.1e+47], N[Not[LessEqual[q$95$m, 2.6e+90]], $MachinePrecision]], (-q$95$m), N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q < 3.1000000000000001e47 or 2.5999999999999998e90 < q

   1. Initial program 25.9%

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

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

      2. lower-neg.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if 3.1000000000000001e47 < q < 2.5999999999999998e90

   1. Initial program 2.2%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in r around 0

      \[\leadsto -1 \cdot \frac{{q}^{2}}{\color{blue}{r}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(-q\right) \cdot q}{r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{elif}\;q \leq 3.1 \cdot 10^{+47} \lor \neg \left(q \leq 2.6 \cdot 10^{+90}\right):\\ \;\;\;\;-q\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.3% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \left(p + \left(\left|p\right| + \left(\left|r\right| - r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 9e-56) (* 0.5 (+ p (+ (fabs p) (- (fabs r) r)))) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (fabs(p) + (fabs(r) - r)));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 9d-56) then
        tmp = 0.5d0 * (p + (abs(p) + (abs(r) - r)))
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9e-56) {
		tmp = 0.5 * (p + (Math.abs(p) + (Math.abs(r) - r)));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 9e-56:
		tmp = 0.5 * (p + (math.fabs(p) + (math.fabs(r) - r)))
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 9e-56)
		tmp = Float64(0.5 * Float64(p + Float64(abs(p) + Float64(abs(r) - r))));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 9e-56)
		tmp = 0.5 * (p + (abs(p) + (abs(r) - r)));
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 9e-56], N[(0.5 * N[(p + N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 9.0000000000000001e-56

   1. Initial program 29.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-fabs.f64 18.3

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 18.3%

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

   if 9.0000000000000001e-56 < q

   1. Initial program 22.0%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.0

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

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{-q} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 36.0% accurate, 83.3× speedup

Math

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

FPCore

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

C

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

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = -q_m
end function

Java

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

Python

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

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-q_m)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in q around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 16.6

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

5. Applied rewrites 16.6%

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

Reproduce

herbie shell --seed 2024343 
(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)))))))