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

Percentage Accurate: 44.7% → 79.5%

Time: 6.6s

Alternatives: 11

Speedup: 16.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 44.7% 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: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{if}\;q\_m \leq 64000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;q\_m \leq 8 \cdot 10^{+87}:\\ \;\;\;\;{2}^{-1} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q\_m}^{2}}\right)\\ \mathbf{elif}\;q\_m \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p}{q\_m} \cdot \frac{p}{q\_m}, 0.125, \mathsf{fma}\left(\frac{r + \left|p\right|}{q\_m}, 0.5, 1\right)\right) \cdot q\_m\\ \end{array} \end{array} \]

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 (- p (fabs p)) -0.5 r)))
   (if (<= q_m 64000.0)
     t_0
     (if (<= q_m 8e+87)
       (*
        (pow 2.0 -1.0)
        (+
         (+ (fabs p) (fabs r))
         (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q_m 2.0))))))
       (if (<= q_m 4.5e+160)
         t_0
         (*
          (fma
           (* (/ p q_m) (/ p q_m))
           0.125
           (fma (/ (+ r (fabs p)) q_m) 0.5 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((p - fabs(p)), -0.5, r);
	double tmp;
	if (q_m <= 64000.0) {
		tmp = t_0;
	} else if (q_m <= 8e+87) {
		tmp = pow(2.0, -1.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q_m, 2.0)))));
	} else if (q_m <= 4.5e+160) {
		tmp = t_0;
	} else {
		tmp = fma(((p / q_m) * (p / q_m)), 0.125, fma(((r + fabs(p)) / q_m), 0.5, 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(p - abs(p)), -0.5, r)
	tmp = 0.0
	if (q_m <= 64000.0)
		tmp = t_0;
	elseif (q_m <= 8e+87)
		tmp = Float64((2.0 ^ -1.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q_m ^ 2.0))))));
	elseif (q_m <= 4.5e+160)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(p / q_m) * Float64(p / q_m)), 0.125, fma(Float64(Float64(r + abs(p)) / q_m), 0.5, 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[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] * -0.5 + r), $MachinePrecision]}, If[LessEqual[q$95$m, 64000.0], t$95$0, If[LessEqual[q$95$m, 8e+87], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 4.5e+160], t$95$0, N[(N[(N[(N[(p / q$95$m), $MachinePrecision] * N[(p / q$95$m), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 64000 or 7.9999999999999997e87 < q < 4.4999999999999998e160

   1. Initial program 48.2%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 38.0%

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

   10. Applied rewrites 37.5%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 37.6%

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

   if 64000 < q < 7.9999999999999997e87

   1. Initial program 79.1%

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

   if 4.4999999999999998e160 < q

   1. Initial program 7.7%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 7.7

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

   5. Applied rewrites 7.7%

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 84.5%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 64000:\\ \;\;\;\;\mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{elif}\;q \leq 8 \cdot 10^{+87}:\\ \;\;\;\;{2}^{-1} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right)\\ \mathbf{elif}\;q \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p}{q} \cdot \frac{p}{q}, 0.125, \mathsf{fma}\left(\frac{r + \left|p\right|}{q}, 0.5, 1\right)\right) \cdot q\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{if}\;q\_m \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;q\_m \leq 8 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)} + r\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p}{q\_m} \cdot \frac{p}{q\_m}, 0.125, \mathsf{fma}\left(\frac{r + \left|p\right|}{q\_m}, 0.5, 1\right)\right) \cdot q\_m\\ \end{array} \end{array} \]

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 (- p (fabs p)) -0.5 r)))
   (if (<= q_m 2.7e+18)
     t_0
     (if (<= q_m 8e+87)
       (* (+ (+ (sqrt (fma (* q_m q_m) 4.0 (* p p))) r) (fabs p)) 0.5)
       (if (<= q_m 4.5e+160)
         t_0
         (*
          (fma
           (* (/ p q_m) (/ p q_m))
           0.125
           (fma (/ (+ r (fabs p)) q_m) 0.5 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((p - fabs(p)), -0.5, r);
	double tmp;
	if (q_m <= 2.7e+18) {
		tmp = t_0;
	} else if (q_m <= 8e+87) {
		tmp = ((sqrt(fma((q_m * q_m), 4.0, (p * p))) + r) + fabs(p)) * 0.5;
	} else if (q_m <= 4.5e+160) {
		tmp = t_0;
	} else {
		tmp = fma(((p / q_m) * (p / q_m)), 0.125, fma(((r + fabs(p)) / q_m), 0.5, 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(p - abs(p)), -0.5, r)
	tmp = 0.0
	if (q_m <= 2.7e+18)
		tmp = t_0;
	elseif (q_m <= 8e+87)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(q_m * q_m), 4.0, Float64(p * p))) + r) + abs(p)) * 0.5);
	elseif (q_m <= 4.5e+160)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(p / q_m) * Float64(p / q_m)), 0.125, fma(Float64(Float64(r + abs(p)) / q_m), 0.5, 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[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] * -0.5 + r), $MachinePrecision]}, If[LessEqual[q$95$m, 2.7e+18], t$95$0, If[LessEqual[q$95$m, 8e+87], N[(N[(N[(N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 4.5e+160], t$95$0, N[(N[(N[(N[(p / q$95$m), $MachinePrecision] * N[(p / q$95$m), $MachinePrecision]), $MachinePrecision] * 0.125 + N[(N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.7e18 or 7.9999999999999997e87 < q < 4.4999999999999998e160

   1. Initial program 48.5%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 38.0%

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

   10. Applied rewrites 37.5%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 37.6%

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

   if 2.7e18 < q < 7.9999999999999997e87

   1. Initial program 81.2%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 61.8%

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

   if 4.4999999999999998e160 < q

   1. Initial program 7.7%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 7.7

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

   5. Applied rewrites 7.7%

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 84.5%

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

Alternative 3: 79.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{if}\;q\_m \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;q\_m \leq 8 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)} + r\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (fma (- p (fabs p)) -0.5 r)))
   (if (<= q_m 2.7e+18)
     t_0
     (if (<= q_m 8e+87)
       (* (+ (+ (sqrt (fma (* q_m q_m) 4.0 (* p p))) r) (fabs p)) 0.5)
       (if (<= q_m 4.5e+160) t_0 (fma 0.5 (+ 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 t_0 = fma((p - fabs(p)), -0.5, r);
	double tmp;
	if (q_m <= 2.7e+18) {
		tmp = t_0;
	} else if (q_m <= 8e+87) {
		tmp = ((sqrt(fma((q_m * q_m), 4.0, (p * p))) + r) + fabs(p)) * 0.5;
	} else if (q_m <= 4.5e+160) {
		tmp = t_0;
	} else {
		tmp = fma(0.5, (r + fabs(p)), 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(p - abs(p)), -0.5, r)
	tmp = 0.0
	if (q_m <= 2.7e+18)
		tmp = t_0;
	elseif (q_m <= 8e+87)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(q_m * q_m), 4.0, Float64(p * p))) + r) + abs(p)) * 0.5);
	elseif (q_m <= 4.5e+160)
		tmp = t_0;
	else
		tmp = fma(0.5, Float64(r + abs(p)), 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[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] * -0.5 + r), $MachinePrecision]}, If[LessEqual[q$95$m, 2.7e+18], t$95$0, If[LessEqual[q$95$m, 8e+87], N[(N[(N[(N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 4.5e+160], t$95$0, N[(0.5 * N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.7e18 or 7.9999999999999997e87 < q < 4.4999999999999998e160

   1. Initial program 48.5%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 38.0%

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

   10. Applied rewrites 37.5%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 37.6%

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

   if 2.7e18 < q < 7.9999999999999997e87

   1. Initial program 81.2%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 61.8%

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

   if 4.4999999999999998e160 < q

   1. Initial program 7.7%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 85.3%

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

Alternative 4: 81.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := r + \left|p\right|\\ \mathbf{if}\;q\_m \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;-0.5 \cdot \left(\left(p - t\_0\right) - \left|r\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ r (fabs p))))
   (if (<= q_m 4.5e+160) (* -0.5 (- (- p t_0) (fabs r))) (fma 0.5 t_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 = r + fabs(p);
	double tmp;
	if (q_m <= 4.5e+160) {
		tmp = -0.5 * ((p - t_0) - fabs(r));
	} else {
		tmp = fma(0.5, t_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(r + abs(p))
	tmp = 0.0
	if (q_m <= 4.5e+160)
		tmp = Float64(-0.5 * Float64(Float64(p - t_0) - abs(r)));
	else
		tmp = fma(0.5, t_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[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 4.5e+160], N[(-0.5 * N[(N[(p - t$95$0), $MachinePrecision] - N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0 + q$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 4.4999999999999998e160

   1. Initial program 50.7%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 36.6%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 36.9%

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

   if 4.4999999999999998e160 < q

   1. Initial program 7.7%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 85.3%

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

Alternative 5: 63.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -6.8 \cdot 10^{+77}:\\ \;\;\;\;\left(p - \left|p\right|\right) \cdot -0.5\\ \mathbf{elif}\;p \leq 3.6 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|p\right|, r\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -6.8e+77)
   (* (- p (fabs p)) -0.5)
   (if (<= p 3.6e-285) (fma 0.5 r q_m) (fma 0.5 (fabs p) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -6.8e+77) {
		tmp = (p - fabs(p)) * -0.5;
	} else if (p <= 3.6e-285) {
		tmp = fma(0.5, r, q_m);
	} else {
		tmp = fma(0.5, fabs(p), r);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -6.8e+77)
		tmp = Float64(Float64(p - abs(p)) * -0.5);
	elseif (p <= 3.6e-285)
		tmp = fma(0.5, r, q_m);
	else
		tmp = fma(0.5, abs(p), r);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -6.8e+77], N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[p, 3.6e-285], N[(0.5 * r + q$95$m), $MachinePrecision], N[(0.5 * N[Abs[p], $MachinePrecision] + r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < -6.79999999999999993e77

   1. Initial program 30.5%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 84.4%

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

   10. Applied rewrites 84.0%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 74.9%

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

   if -6.79999999999999993e77 < p < 3.60000000000000004e-285

   1. Initial program 57.4%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 40.6%

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

   10. Applied rewrites 35.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 35.7%

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

   if 3.60000000000000004e-285 < p

   1. Initial program 44.5%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 14.1

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

   5. Applied rewrites 14.1%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 18.4%

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

   10. Applied rewrites 18.0%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 24.6%

       \[\leadsto \mathsf{fma}\left(0.5, \left|p\right|, r\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.9% accurate, 13.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 4.5e+160)
   (fma (- p (fabs p)) -0.5 r)
   (fma 0.5 (+ 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 (q_m <= 4.5e+160) {
		tmp = fma((p - fabs(p)), -0.5, r);
	} else {
		tmp = fma(0.5, (r + fabs(p)), 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 <= 4.5e+160)
		tmp = fma(Float64(p - abs(p)), -0.5, r);
	else
		tmp = fma(0.5, Float64(r + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 4.5e+160], N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] * -0.5 + r), $MachinePrecision], N[(0.5 * N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 4.4999999999999998e160

   1. Initial program 50.7%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 36.6%

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 36.2%

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

   if 4.4999999999999998e160 < q

   1. Initial program 7.7%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 85.3%

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

Alternative 7: 80.5% accurate, 13.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(p - \left|p\right|, -0.5, r\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 4.5e+160) (fma (- p (fabs p)) -0.5 r) (fma 0.5 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 <= 4.5e+160) {
		tmp = fma((p - fabs(p)), -0.5, r);
	} else {
		tmp = fma(0.5, r, 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 <= 4.5e+160)
		tmp = fma(Float64(p - abs(p)), -0.5, r);
	else
		tmp = fma(0.5, r, q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 4.5e+160], N[(N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision] * -0.5 + r), $MachinePrecision], N[(0.5 * r + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 4.4999999999999998e160

   1. Initial program 50.7%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 36.6%

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 36.2%

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

   if 4.4999999999999998e160 < q

   1. Initial program 7.7%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 84.6%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 84.5%

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

Alternative 8: 55.4% accurate, 16.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|p\right|, r\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 6.5e-15) (fma 0.5 r q_m) (fma 0.5 (fabs p) r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 6.5e-15) {
		tmp = fma(0.5, r, q_m);
	} else {
		tmp = fma(0.5, fabs(p), r);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 6.5e-15)
		tmp = fma(0.5, r, q_m);
	else
		tmp = fma(0.5, abs(p), r);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 6.5e-15], N[(0.5 * r + q$95$m), $MachinePrecision], N[(0.5 * N[Abs[p], $MachinePrecision] + r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.49999999999999991e-15

   1. Initial program 49.6%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.1

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

   5. Applied rewrites 28.1%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 30.3%

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

   10. Applied rewrites 23.2%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 19.9%

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

   if 6.49999999999999991e-15 < r

   1. Initial program 33.6%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 71.8%

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

   10. Applied rewrites 71.8%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 63.9%

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

Alternative 9: 41.4% accurate, 19.2× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -2.2 \cdot 10^{+204}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -2.2e+204) (* -0.5 p) (fma 0.5 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 (p <= -2.2e+204) {
		tmp = -0.5 * p;
	} else {
		tmp = fma(0.5, r, 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 (p <= -2.2e+204)
		tmp = Float64(-0.5 * p);
	else
		tmp = fma(0.5, r, q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -2.2e+204], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -2.20000000000000011e204

   1. Initial program 7.9%

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

   3. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
   4. Step-by-step derivation

      1. lower-*.f64 17.7

         \[\leadsto \color{blue}{-0.5 \cdot p} \]

   5. Applied rewrites 17.7%

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

   if -2.20000000000000011e204 < p

   1. Initial program 49.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 31.4%

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

   10. Applied rewrites 26.5%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 23.5%

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

Alternative 10: 36.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -2.2 \cdot 10^{+204}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;q\_m\\ \end{array} \end{array} \]

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 (<= p -2.2e+204) (* -0.5 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 (p <= -2.2e+204) {
		tmp = -0.5 * 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 (p <= (-2.2d+204)) then
        tmp = (-0.5d0) * 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 (p <= -2.2e+204) {
		tmp = -0.5 * 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 p <= -2.2e+204:
		tmp = -0.5 * 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 (p <= -2.2e+204)
		tmp = Float64(-0.5 * p);
	else
		tmp = 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 (p <= -2.2e+204)
		tmp = -0.5 * 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[p, -2.2e+204], N[(-0.5 * p), $MachinePrecision], q$95$m]

Derivation

1. Split input into 2 regimes

2. if p < -2.20000000000000011e204

   1. Initial program 7.9%

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

   3. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
   4. Step-by-step derivation

      1. lower-*.f64 17.7

         \[\leadsto \color{blue}{-0.5 \cdot p} \]

   5. Applied rewrites 17.7%

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

   if -2.20000000000000011e204 < p

   1. Initial program 49.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 31.4%

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

   10. Applied rewrites 21.8%

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

   11. Taylor expanded in q around -inf

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

   13. Applied rewrites 20.9%

       \[\leadsto q \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -2.2 \cdot 10^{+204}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;q\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 250.0× speedup

Math

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

FPCore

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

C

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

Fortran

q_m = 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 q_m
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

3. Taylor expanded in q around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-fabs.f64 27.9

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

5. Applied rewrites 27.9%

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

6. Taylor expanded in p around 0

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

8. Applied rewrites 30.3%

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

10. Applied rewrites 20.0%

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

11. Taylor expanded in q around -inf

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

13. Applied rewrites 19.3%

    \[\leadsto q \]

14. Final simplification 19.3%

    \[\leadsto q \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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)))))))