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

Percentage Accurate: 24.7% → 68.4%

Time: 10.2s

Alternatives: 6

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: 24.7% 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: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(2 \cdot q\_m, p\right)\\ t_1 := {t\_0}^{0.5}\\ t_2 := \frac{p}{t\_1}\\ \mathbf{if}\;r \leq 1.15 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{t\_1}, t\_2, \left(\left|p\right| - \mathsf{hypot}\left(p, q\_m \cdot 2\right)\right) + \left|r\right|\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{\mathsf{fma}\left(0.0625 \cdot {4}^{0.25}, \sqrt{{\left({q\_m}^{3}\right)}^{-1}} \cdot \left(p \cdot p\right), \sqrt{q\_m} \cdot {4}^{0.25}\right)}, t\_2, \left(\left|p\right| + \left|r\right|\right) - t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(p, \frac{\frac{p}{r \cdot r} + {r}^{-1}}{r \cdot r}, {\left(r \cdot r\right)}^{-1}\right), p, \frac{-1}{r}\right), q\_m \cdot q\_m, 0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (hypot (* 2.0 q_m) p)) (t_1 (pow t_0 0.5)) (t_2 (/ p t_1)))
   (if (<= r 1.15e-291)
     (*
      (fma (/ r t_1) t_2 (+ (- (fabs p) (hypot p (* q_m 2.0))) (fabs r)))
      0.5)
     (if (<= r 1.15e+38)
       (*
        (fma
         (/
          r
          (fma
           (* 0.0625 (pow 4.0 0.25))
           (* (sqrt (pow (pow q_m 3.0) -1.0)) (* p p))
           (* (sqrt q_m) (pow 4.0 0.25))))
         t_2
         (- (+ (fabs p) (fabs r)) t_0))
        0.5)
       (fma
        (fma
         (-
          (fma
           p
           (/ (+ (/ p (* r r)) (pow r -1.0)) (* r r))
           (pow (* r r) -1.0)))
         p
         (/ -1.0 r))
        (* q_m q_m)
        (* 0.5 (+ (+ p (- (fabs r) r)) (fabs p))))))))

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 t_1 = pow(t_0, 0.5);
	double t_2 = p / t_1;
	double tmp;
	if (r <= 1.15e-291) {
		tmp = fma((r / t_1), t_2, ((fabs(p) - hypot(p, (q_m * 2.0))) + fabs(r))) * 0.5;
	} else if (r <= 1.15e+38) {
		tmp = fma((r / fma((0.0625 * pow(4.0, 0.25)), (sqrt(pow(pow(q_m, 3.0), -1.0)) * (p * p)), (sqrt(q_m) * pow(4.0, 0.25)))), t_2, ((fabs(p) + fabs(r)) - t_0)) * 0.5;
	} else {
		tmp = fma(fma(-fma(p, (((p / (r * r)) + pow(r, -1.0)) / (r * r)), pow((r * r), -1.0)), p, (-1.0 / r)), (q_m * q_m), (0.5 * ((p + (fabs(r) - r)) + fabs(p))));
	}
	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)
	t_1 = t_0 ^ 0.5
	t_2 = Float64(p / t_1)
	tmp = 0.0
	if (r <= 1.15e-291)
		tmp = Float64(fma(Float64(r / t_1), t_2, Float64(Float64(abs(p) - hypot(p, Float64(q_m * 2.0))) + abs(r))) * 0.5);
	elseif (r <= 1.15e+38)
		tmp = Float64(fma(Float64(r / fma(Float64(0.0625 * (4.0 ^ 0.25)), Float64(sqrt(((q_m ^ 3.0) ^ -1.0)) * Float64(p * p)), Float64(sqrt(q_m) * (4.0 ^ 0.25)))), t_2, Float64(Float64(abs(p) + abs(r)) - t_0)) * 0.5);
	else
		tmp = fma(fma(Float64(-fma(p, Float64(Float64(Float64(p / Float64(r * r)) + (r ^ -1.0)) / Float64(r * r)), (Float64(r * r) ^ -1.0))), p, Float64(-1.0 / r)), Float64(q_m * q_m), Float64(0.5 * Float64(Float64(p + Float64(abs(r) - r)) + abs(p))));
	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]}, Block[{t$95$1 = N[Power[t$95$0, 0.5], $MachinePrecision]}, Block[{t$95$2 = N[(p / t$95$1), $MachinePrecision]}, If[LessEqual[r, 1.15e-291], N[(N[(N[(r / t$95$1), $MachinePrecision] * t$95$2 + N[(N[(N[Abs[p], $MachinePrecision] - N[Sqrt[p ^ 2 + N[(q$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 1.15e+38], N[(N[(N[(r / N[(N[(0.0625 * N[Power[4.0, 0.25], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[Power[N[Power[q$95$m, 3.0], $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(p * p), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[q$95$m], $MachinePrecision] * N[Power[4.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-N[(p * N[(N[(N[(p / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[Power[r, -1.0], $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[Power[N[(r * r), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]) * p + N[(-1.0 / r), $MachinePrecision]), $MachinePrecision] * N[(q$95$m * q$95$m), $MachinePrecision] + N[(0.5 * N[(N[(p + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.15e-291

   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 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 19.6%

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

   7. Applied rewrites 22.6%

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

   9. Applied rewrites 38.0%

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

   11. Applied rewrites 45.2%

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

   if 1.15e-291 < r < 1.1500000000000001e38

   1. Initial program 39.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 32.1%

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

   7. Applied rewrites 34.5%

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

   9. Applied rewrites 52.7%

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

   10. Taylor expanded in p around 0

       \[\leadsto \mathsf{fma}\left(\frac{r}{\frac{1}{16} \cdot \left({4}^{\frac{1}{4}} \cdot \left({p}^{2} \cdot \sqrt{\frac{1}{{q}^{3}}}\right)\right) + {4}^{\frac{1}{4}} \cdot \sqrt{q}}, \frac{p}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{\frac{1}{2}}}, \left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(2 \cdot q, p\right)\right) \cdot \frac{1}{2} \]
   11. Step-by-step derivation

   12. Applied rewrites 36.8%

       \[\leadsto \mathsf{fma}\left(\frac{r}{\mathsf{fma}\left(0.0625 \cdot {4}^{0.25}, \sqrt{\frac{1}{{q}^{3}}} \cdot \left(p \cdot p\right), \sqrt{q} \cdot {4}^{0.25}\right)}, \frac{p}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{0.5}}, \left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(2 \cdot q, p\right)\right) \cdot 0.5 \]

   if 1.1500000000000001e38 < r

   1. Initial program 4.9%

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

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right) + p \cdot \left(\frac{1}{2} \cdot \left(r \cdot \sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}}\right) + p \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}} \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right) + \frac{-1}{4} \cdot \left(\left(p \cdot \left(r \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{{\left(4 \cdot {q}^{2} + {r}^{2}\right)}^{3}}}\right)\right)\right)} \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot p, \mathsf{fma}\left(-r, \frac{r}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}, 1\right) \cdot \mathsf{fma}\left(\sqrt{\frac{1}{{\left(\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)\right)}^{3}}}, r \cdot p, \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), \left(0.5 \cdot r\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), p, \left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5\right)} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 62.7%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{0.5}}, \frac{p}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{0.5}}, \left(\left|p\right| - \mathsf{hypot}\left(p, q \cdot 2\right)\right) + \left|r\right|\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{\mathsf{fma}\left(0.0625 \cdot {4}^{0.25}, \sqrt{{\left({q}^{3}\right)}^{-1}} \cdot \left(p \cdot p\right), \sqrt{q} \cdot {4}^{0.25}\right)}, \frac{p}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{0.5}}, \left(\left|p\right| + \left|r\right|\right) - \mathsf{hypot}\left(2 \cdot q, p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(p, \frac{\frac{p}{r \cdot r} + {r}^{-1}}{r \cdot r}, {\left(r \cdot r\right)}^{-1}\right), p, \frac{-1}{r}\right), q \cdot q, 0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(2 \cdot q\_m, p\right)\right)}^{0.5}\\ \mathbf{if}\;r \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{t\_0}, \frac{p}{t\_0}, \left(\left|p\right| - \mathsf{hypot}\left(p, q\_m \cdot 2\right)\right) + \left|r\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(p, \frac{\frac{p}{r \cdot r} + {r}^{-1}}{r \cdot r}, {\left(r \cdot r\right)}^{-1}\right), p, \frac{-1}{r}\right), q\_m \cdot q\_m, 0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (pow (hypot (* 2.0 q_m) p) 0.5)))
   (if (<= r 1.8e+38)
     (*
      (fma (/ r t_0) (/ p t_0) (+ (- (fabs p) (hypot p (* q_m 2.0))) (fabs r)))
      0.5)
     (fma
      (fma
       (-
        (fma p (/ (+ (/ p (* r r)) (pow r -1.0)) (* r r)) (pow (* r r) -1.0)))
       p
       (/ -1.0 r))
      (* q_m q_m)
      (* 0.5 (+ (+ p (- (fabs r) r)) (fabs p)))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = pow(hypot((2.0 * q_m), p), 0.5);
	double tmp;
	if (r <= 1.8e+38) {
		tmp = fma((r / t_0), (p / t_0), ((fabs(p) - hypot(p, (q_m * 2.0))) + fabs(r))) * 0.5;
	} else {
		tmp = fma(fma(-fma(p, (((p / (r * r)) + pow(r, -1.0)) / (r * r)), pow((r * r), -1.0)), p, (-1.0 / r)), (q_m * q_m), (0.5 * ((p + (fabs(r) - r)) + fabs(p))));
	}
	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) ^ 0.5
	tmp = 0.0
	if (r <= 1.8e+38)
		tmp = Float64(fma(Float64(r / t_0), Float64(p / t_0), Float64(Float64(abs(p) - hypot(p, Float64(q_m * 2.0))) + abs(r))) * 0.5);
	else
		tmp = fma(fma(Float64(-fma(p, Float64(Float64(Float64(p / Float64(r * r)) + (r ^ -1.0)) / Float64(r * r)), (Float64(r * r) ^ -1.0))), p, Float64(-1.0 / r)), Float64(q_m * q_m), Float64(0.5 * Float64(Float64(p + Float64(abs(r) - r)) + abs(p))));
	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[Power[N[Sqrt[N[(2.0 * q$95$m), $MachinePrecision] ^ 2 + p ^ 2], $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[r, 1.8e+38], N[(N[(N[(r / t$95$0), $MachinePrecision] * N[(p / t$95$0), $MachinePrecision] + N[(N[(N[Abs[p], $MachinePrecision] - N[Sqrt[p ^ 2 + N[(q$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-N[(p * N[(N[(N[(p / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[Power[r, -1.0], $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[Power[N[(r * r), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]) * p + N[(-1.0 / r), $MachinePrecision]), $MachinePrecision] * N[(q$95$m * q$95$m), $MachinePrecision] + N[(0.5 * N[(N[(p + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.79999999999999985e38

   1. Initial program 31.5%

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

   3. Taylor expanded in r around 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 23.7%

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

   7. Applied rewrites 26.5%

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

   9. Applied rewrites 42.7%

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

   11. Applied rewrites 48.9%

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

   if 1.79999999999999985e38 < r

   1. Initial program 4.9%

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

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right) + p \cdot \left(\frac{1}{2} \cdot \left(r \cdot \sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}}\right) + p \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}} \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right) + \frac{-1}{4} \cdot \left(\left(p \cdot \left(r \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{{\left(4 \cdot {q}^{2} + {r}^{2}\right)}^{3}}}\right)\right)\right)} \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot p, \mathsf{fma}\left(-r, \frac{r}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}, 1\right) \cdot \mathsf{fma}\left(\sqrt{\frac{1}{{\left(\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)\right)}^{3}}}, r \cdot p, \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), \left(0.5 \cdot r\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), p, \left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5\right)} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 62.7%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{0.5}}, \frac{p}{{\left(\mathsf{hypot}\left(2 \cdot q, p\right)\right)}^{0.5}}, \left(\left|p\right| - \mathsf{hypot}\left(p, q \cdot 2\right)\right) + \left|r\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(p, \frac{\frac{p}{r \cdot r} + {r}^{-1}}{r \cdot r}, {\left(r \cdot r\right)}^{-1}\right), p, \frac{-1}{r}\right), q \cdot q, 0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(p, q\_m \cdot 2\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;r \leq -2.35 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + p\right) + \left|r\right|, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{t\_1}, \frac{p}{t\_1}, \left(\left|r\right| + \left|p\right|\right) - t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(p, \frac{\frac{p}{r \cdot r} + {r}^{-1}}{r \cdot r}, {\left(r \cdot r\right)}^{-1}\right), p, \frac{-1}{r}\right), q\_m \cdot q\_m, 0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (hypot p (* q_m 2.0))) (t_1 (sqrt t_0)))
   (if (<= r -2.35e-6)
     (* (fma (+ (+ (fabs p) p) (fabs r)) (/ 0.5 r) -0.5) r)
     (if (<= r 1.15e+38)
       (* (fma (/ r t_1) (/ p t_1) (- (+ (fabs r) (fabs p)) t_0)) 0.5)
       (fma
        (fma
         (-
          (fma
           p
           (/ (+ (/ p (* r r)) (pow r -1.0)) (* r r))
           (pow (* r r) -1.0)))
         p
         (/ -1.0 r))
        (* q_m q_m)
        (* 0.5 (+ (+ p (- (fabs r) r)) (fabs p))))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = hypot(p, (q_m * 2.0));
	double t_1 = sqrt(t_0);
	double tmp;
	if (r <= -2.35e-6) {
		tmp = fma(((fabs(p) + p) + fabs(r)), (0.5 / r), -0.5) * r;
	} else if (r <= 1.15e+38) {
		tmp = fma((r / t_1), (p / t_1), ((fabs(r) + fabs(p)) - t_0)) * 0.5;
	} else {
		tmp = fma(fma(-fma(p, (((p / (r * r)) + pow(r, -1.0)) / (r * r)), pow((r * r), -1.0)), p, (-1.0 / r)), (q_m * q_m), (0.5 * ((p + (fabs(r) - r)) + fabs(p))));
	}
	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(p, Float64(q_m * 2.0))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (r <= -2.35e-6)
		tmp = Float64(fma(Float64(Float64(abs(p) + p) + abs(r)), Float64(0.5 / r), -0.5) * r);
	elseif (r <= 1.15e+38)
		tmp = Float64(fma(Float64(r / t_1), Float64(p / t_1), Float64(Float64(abs(r) + abs(p)) - t_0)) * 0.5);
	else
		tmp = fma(fma(Float64(-fma(p, Float64(Float64(Float64(p / Float64(r * r)) + (r ^ -1.0)) / Float64(r * r)), (Float64(r * r) ^ -1.0))), p, Float64(-1.0 / r)), Float64(q_m * q_m), Float64(0.5 * Float64(Float64(p + Float64(abs(r) - r)) + abs(p))));
	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[p ^ 2 + N[(q$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[r, -2.35e-6], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(0.5 / r), $MachinePrecision] + -0.5), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[r, 1.15e+38], N[(N[(N[(r / t$95$1), $MachinePrecision] * N[(p / t$95$1), $MachinePrecision] + N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-N[(p * N[(N[(N[(p / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[Power[r, -1.0], $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[Power[N[(r * r), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]) * p + N[(-1.0 / r), $MachinePrecision]), $MachinePrecision] * N[(q$95$m * q$95$m), $MachinePrecision] + N[(0.5 * N[(N[(p + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.34999999999999995e-6

   1. Initial program 17.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 3.3%

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

   7. Applied rewrites 10.1%

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

   if -2.34999999999999995e-6 < r < 1.1500000000000001e38

   1. Initial program 39.8%

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

   3. Taylor expanded in r around 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 33.2%

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

   7. Applied rewrites 36.9%

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

   9. Applied rewrites 51.2%

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

   11. Applied rewrites 51.2%

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

   if 1.1500000000000001e38 < r

   1. Initial program 4.9%

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

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right) + p \cdot \left(\frac{1}{2} \cdot \left(r \cdot \sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}}\right) + p \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}} \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right) + \frac{-1}{4} \cdot \left(\left(p \cdot \left(r \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{{\left(4 \cdot {q}^{2} + {r}^{2}\right)}^{3}}}\right)\right)\right)} \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot p, \mathsf{fma}\left(-r, \frac{r}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}, 1\right) \cdot \mathsf{fma}\left(\sqrt{\frac{1}{{\left(\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)\right)}^{3}}}, r \cdot p, \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), \left(0.5 \cdot r\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), p, \left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5\right)} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 62.7%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.35 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + p\right) + \left|r\right|, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{\sqrt{\mathsf{hypot}\left(p, q \cdot 2\right)}}, \frac{p}{\sqrt{\mathsf{hypot}\left(p, q \cdot 2\right)}}, \left(\left|r\right| + \left|p\right|\right) - \mathsf{hypot}\left(p, q \cdot 2\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(p, \frac{\frac{p}{r \cdot r} + {r}^{-1}}{r \cdot r}, {\left(r \cdot r\right)}^{-1}\right), p, \frac{-1}{r}\right), q \cdot q, 0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{-198}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \left(\left|r\right| - r\right)\right) + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 2e-198)
   (* 0.5 (+ (+ p (- (fabs r) r)) (fabs p)))
   (- 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 (pow(q_m, 2.0) <= 2e-198) {
		tmp = 0.5 * ((p + (fabs(r) - r)) + fabs(p));
	} 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 ** 2.0d0) <= 2d-198) then
        tmp = 0.5d0 * ((p + (abs(r) - r)) + abs(p))
    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 (Math.pow(q_m, 2.0) <= 2e-198) {
		tmp = 0.5 * ((p + (Math.abs(r) - r)) + Math.abs(p));
	} 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 math.pow(q_m, 2.0) <= 2e-198:
		tmp = 0.5 * ((p + (math.fabs(r) - r)) + math.fabs(p))
	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 ^ 2.0) <= 2e-198)
		tmp = Float64(0.5 * Float64(Float64(p + Float64(abs(r) - r)) + abs(p)));
	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 ^ 2.0) <= 2e-198)
		tmp = 0.5 * ((p + (abs(r) - r)) + abs(p));
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 2e-198], N[(0.5 * N[(N[(p + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.9999999999999998e-198

   1. Initial program 21.5%

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

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{4 \cdot {q}^{2} + {r}^{2}}\right) + p \cdot \left(\frac{1}{2} \cdot \left(r \cdot \sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}}\right) + p \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{4 \cdot {q}^{2} + {r}^{2}}} \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right) + \frac{-1}{4} \cdot \left(\left(p \cdot \left(r \cdot \left(1 - \frac{{r}^{2}}{4 \cdot {q}^{2} + {r}^{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{{\left(4 \cdot {q}^{2} + {r}^{2}\right)}^{3}}}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot p, \mathsf{fma}\left(-r, \frac{r}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}, 1\right) \cdot \mathsf{fma}\left(\sqrt{\frac{1}{{\left(\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)\right)}^{3}}}, r \cdot p, \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), \left(0.5 \cdot r\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}\right), p, \left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5\right)} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 40.5%

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

   if 1.9999999999999998e-198 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.2%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.5

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

   5. Applied rewrites 29.5%

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

Alternative 5: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{-198}:\\ \;\;\;\;\left(\left(p + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 2e-198) (* (+ (+ p (fabs r)) (fabs p)) 0.5) (- 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 (pow(q_m, 2.0) <= 2e-198) {
		tmp = ((p + fabs(r)) + fabs(p)) * 0.5;
	} 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 ** 2.0d0) <= 2d-198) then
        tmp = ((p + abs(r)) + abs(p)) * 0.5d0
    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 (Math.pow(q_m, 2.0) <= 2e-198) {
		tmp = ((p + Math.abs(r)) + Math.abs(p)) * 0.5;
	} 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 math.pow(q_m, 2.0) <= 2e-198:
		tmp = ((p + math.fabs(r)) + math.fabs(p)) * 0.5
	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 ^ 2.0) <= 2e-198)
		tmp = Float64(Float64(Float64(p + abs(r)) + abs(p)) * 0.5);
	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 ^ 2.0) <= 2e-198)
		tmp = ((p + abs(r)) + abs(p)) * 0.5;
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 2e-198], N[(N[(N[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.9999999999999998e-198

   1. Initial program 21.5%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 13.0%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 16.5%

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

   if 1.9999999999999998e-198 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.2%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.5

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

   5. Applied rewrites 29.5%

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

Alternative 6: 35.6% 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 25.1%

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

3. Taylor expanded in 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 21.1

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

5. Applied rewrites 21.1%

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

Reproduce

herbie shell --seed 2024318 
(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)))))))