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

Percentage Accurate: 44.4% → 82.4%

Time: 7.4s

Alternatives: 13

Speedup: 11.4×

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: 44.4% 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: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|r\right| - p\right) + r, 0.5, \left|p\right| \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{q\_m}, 0.5, 1\right) \cdot q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e+232)
   (fma (+ (- (fabs r) p) r) 0.5 (* (fabs p) 0.5))
   (* (fma (/ (+ (fabs 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 tmp;
	if (pow(q_m, 2.0) <= 2e+232) {
		tmp = fma(((fabs(r) - p) + r), 0.5, (fabs(p) * 0.5));
	} else {
		tmp = fma(((fabs(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)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = fma(Float64(Float64(abs(r) - p) + r), 0.5, Float64(abs(p) * 0.5));
	else
		tmp = Float64(fma(Float64(Float64(abs(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_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+232], N[(N[(N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision] + r), $MachinePrecision] * 0.5 + N[(N[Abs[p], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.00000000000000011e232

   1. Initial program 55.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 44.7%

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

   10. Applied rewrites 44.7%

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

   if 2.00000000000000011e232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 17.8%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 38.0

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

   5. Applied rewrites 38.0%

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

Alternative 2: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|r\right| - p\right) + r, 0.5, \left|p\right| \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e+232)
   (fma (+ (- (fabs r) p) r) 0.5 (* (fabs p) 0.5))
   (* (+ (fma q_m 2.0 (fabs r)) (fabs p)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 2e+232) {
		tmp = fma(((fabs(r) - p) + r), 0.5, (fabs(p) * 0.5));
	} else {
		tmp = (fma(q_m, 2.0, fabs(r)) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = fma(Float64(Float64(abs(r) - p) + r), 0.5, Float64(abs(p) * 0.5));
	else
		tmp = Float64(Float64(fma(q_m, 2.0, abs(r)) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+232], N[(N[(N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision] + r), $MachinePrecision] * 0.5 + N[(N[Abs[p], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(q$95$m * 2.0 + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.00000000000000011e232

   1. Initial program 55.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 44.7%

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

   10. Applied rewrites 44.7%

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

   if 2.00000000000000011e232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 17.8%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 38.0%

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

Alternative 3: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(r, 0.5, \left(\left|r\right| - \left(p - \left|p\right|\right)\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e+232)
   (fma r 0.5 (* (- (fabs r) (- p (fabs p))) 0.5))
   (* (+ (fma q_m 2.0 (fabs r)) (fabs p)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 2e+232) {
		tmp = fma(r, 0.5, ((fabs(r) - (p - fabs(p))) * 0.5));
	} else {
		tmp = (fma(q_m, 2.0, fabs(r)) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = fma(r, 0.5, Float64(Float64(abs(r) - Float64(p - abs(p))) * 0.5));
	else
		tmp = Float64(Float64(fma(q_m, 2.0, abs(r)) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+232], N[(r * 0.5 + N[(N[(N[Abs[r], $MachinePrecision] - N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(q$95$m * 2.0 + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.00000000000000011e232

   1. Initial program 55.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 44.7%

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

   10. Applied rewrites 44.8%

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

   if 2.00000000000000011e232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 17.8%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 38.0%

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

Alternative 4: 82.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\left(\left(\left(r + \left|p\right|\right) + \left|r\right|\right) - p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(q\_m, 2, \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e+232)
   (* (- (+ (+ r (fabs p)) (fabs r)) p) 0.5)
   (* (+ (fma q_m 2.0 (fabs r)) (fabs p)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 2e+232) {
		tmp = (((r + fabs(p)) + fabs(r)) - p) * 0.5;
	} else {
		tmp = (fma(q_m, 2.0, fabs(r)) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = Float64(Float64(Float64(Float64(r + abs(p)) + abs(r)) - p) * 0.5);
	else
		tmp = Float64(Float64(fma(q_m, 2.0, abs(r)) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+232], N[(N[(N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(q$95$m * 2.0 + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.00000000000000011e232

   1. Initial program 55.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 44.7%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 44.5%

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

   if 2.00000000000000011e232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 17.8%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 38.0%

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

Alternative 5: 81.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\left(\left(\left(r + \left|p\right|\right) + \left|r\right|\right) - p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e+232)
   (* (- (+ (+ r (fabs p)) (fabs r)) p) 0.5)
   (* (* q_m 2.0) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 2e+232) {
		tmp = (((r + fabs(p)) + fabs(r)) - p) * 0.5;
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 2d+232) then
        tmp = (((r + abs(p)) + abs(r)) - p) * 0.5d0
    else
        tmp = (q_m * 2.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 2e+232) {
		tmp = (((r + Math.abs(p)) + Math.abs(r)) - p) * 0.5;
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 2e+232:
		tmp = (((r + math.fabs(p)) + math.fabs(r)) - p) * 0.5
	else:
		tmp = (q_m * 2.0) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = Float64(Float64(Float64(Float64(r + abs(p)) + abs(r)) - p) * 0.5);
	else
		tmp = Float64(Float64(q_m * 2.0) * 0.5);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = (((r + abs(p)) + abs(r)) - p) * 0.5;
	else
		tmp = (q_m * 2.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+232], N[(N[(N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(q$95$m * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.00000000000000011e232

   1. Initial program 55.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 44.7%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 44.5%

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

   if 2.00000000000000011e232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 17.8%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 11.2%

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

   9. Taylor expanded in q around inf

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

   11. Applied rewrites 34.5%

       \[\leadsto \left(q \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{+232}:\\ \;\;\;\;0.5 \cdot \left(\left(r + \left(\left|r\right| - p\right)\right) + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e+232)
   (* 0.5 (+ (+ r (- (fabs r) p)) (fabs p)))
   (* (* q_m 2.0) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 2e+232) {
		tmp = 0.5 * ((r + (fabs(r) - p)) + fabs(p));
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 2d+232) then
        tmp = 0.5d0 * ((r + (abs(r) - p)) + abs(p))
    else
        tmp = (q_m * 2.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 2e+232) {
		tmp = 0.5 * ((r + (Math.abs(r) - p)) + Math.abs(p));
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 2e+232:
		tmp = 0.5 * ((r + (math.fabs(r) - p)) + math.fabs(p))
	else:
		tmp = (q_m * 2.0) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = Float64(0.5 * Float64(Float64(r + Float64(abs(r) - p)) + abs(p)));
	else
		tmp = Float64(Float64(q_m * 2.0) * 0.5);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 2e+232)
		tmp = 0.5 * ((r + (abs(r) - p)) + abs(p));
	else
		tmp = (q_m * 2.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+232], N[(0.5 * N[(N[(r + N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(q$95$m * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.00000000000000011e232

   1. Initial program 55.3%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 44.7%

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

   if 2.00000000000000011e232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 17.8%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 11.2%

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

   9. Taylor expanded in q around inf

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

   11. Applied rewrites 34.5%

       \[\leadsto \left(q \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -3.5 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(-p, 0.5, \left|p\right| \cdot 0.5\right)\\ \mathbf{elif}\;r \leq 1.55 \cdot 10^{+41}:\\ \;\;\;\;\left(q\_m \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r -3.5e-176)
   (fma (- p) 0.5 (* (fabs p) 0.5))
   (if (<= r 1.55e+41)
     (* (* q_m 2.0) 0.5)
     (* (+ (+ (fabs r) r) (fabs p)) 0.5))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= -3.5e-176) {
		tmp = fma(-p, 0.5, (fabs(p) * 0.5));
	} else if (r <= 1.55e+41) {
		tmp = (q_m * 2.0) * 0.5;
	} else {
		tmp = ((fabs(r) + r) + fabs(p)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= -3.5e-176)
		tmp = fma(Float64(-p), 0.5, Float64(abs(p) * 0.5));
	elseif (r <= 1.55e+41)
		tmp = Float64(Float64(q_m * 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + abs(p)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, -3.5e-176], N[((-p) * 0.5 + N[(N[Abs[p], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.55e+41], N[(N[(q$95$m * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -3.5e-176

   1. Initial program 43.1%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 12.2

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 14.5%

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

   10. Applied rewrites 14.5%

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

   11. Taylor expanded in p around inf

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

   13. Applied rewrites 13.9%

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

   if -3.5e-176 < r < 1.55e41

   1. Initial program 48.4%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 14.4%

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

   9. Taylor expanded in q around inf

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

   11. Applied rewrites 23.5%

       \[\leadsto \left(q \cdot 2\right) \cdot 0.5 \]

   if 1.55e41 < r

   1. Initial program 29.7%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 26.0

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 71.1%

      \[\leadsto \left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -10000000:\\ \;\;\;\;\mathsf{fma}\left(-p, 0.5, \left|p\right| \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -10000000.0) (fma (- p) 0.5 (* (fabs p) 0.5)) (* (* q_m 2.0) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -10000000.0) {
		tmp = fma(-p, 0.5, (fabs(p) * 0.5));
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -10000000.0)
		tmp = fma(Float64(-p), 0.5, Float64(abs(p) * 0.5));
	else
		tmp = Float64(Float64(q_m * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -10000000.0], N[((-p) * 0.5 + N[(N[Abs[p], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(q$95$m * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -1e7

   1. Initial program 38.8%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 71.3%

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

   10. Applied rewrites 71.3%

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

   11. Taylor expanded in p around inf

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

   13. Applied rewrites 60.9%

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

   if -1e7 < p

   1. Initial program 43.4%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 24.1%

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

   9. Taylor expanded in q around inf

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

   11. Applied rewrites 21.7%

       \[\leadsto \left(q \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -10000000:\\ \;\;\;\;\left(\left(\left|r\right| - p\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -10000000.0)
   (* (+ (- (fabs r) p) (fabs p)) 0.5)
   (* (* q_m 2.0) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -10000000.0) {
		tmp = ((fabs(r) - p) + fabs(p)) * 0.5;
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	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 <= (-10000000.0d0)) then
        tmp = ((abs(r) - p) + abs(p)) * 0.5d0
    else
        tmp = (q_m * 2.0d0) * 0.5d0
    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 <= -10000000.0) {
		tmp = ((Math.abs(r) - p) + Math.abs(p)) * 0.5;
	} else {
		tmp = (q_m * 2.0) * 0.5;
	}
	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 <= -10000000.0:
		tmp = ((math.fabs(r) - p) + math.fabs(p)) * 0.5
	else:
		tmp = (q_m * 2.0) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -10000000.0)
		tmp = Float64(Float64(Float64(abs(r) - p) + abs(p)) * 0.5);
	else
		tmp = Float64(Float64(q_m * 2.0) * 0.5);
	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 <= -10000000.0)
		tmp = ((abs(r) - p) + abs(p)) * 0.5;
	else
		tmp = (q_m * 2.0) * 0.5;
	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, -10000000.0], N[(N[(N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(q$95$m * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -1e7

   1. Initial program 38.8%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 62.7%

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

   if -1e7 < p

   1. Initial program 43.4%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 24.1%

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

   9. Taylor expanded in q around inf

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

   11. Applied rewrites 21.7%

       \[\leadsto \left(q \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 36.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{+190}:\\ \;\;\;\;\left(q\_m \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 2.9e+190) (* (* q_m 2.0) 0.5) (* 0.5 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 <= 2.9e+190) {
		tmp = (q_m * 2.0) * 0.5;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 2.9d+190) then
        tmp = (q_m * 2.0d0) * 0.5d0
    else
        tmp = 0.5d0 * r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 2.9e+190) {
		tmp = (q_m * 2.0) * 0.5;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 2.9e+190:
		tmp = (q_m * 2.0) * 0.5
	else:
		tmp = 0.5 * 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 <= 2.9e+190)
		tmp = Float64(Float64(q_m * 2.0) * 0.5);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 2.9e+190)
		tmp = (q_m * 2.0) * 0.5;
	else
		tmp = 0.5 * r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 2.9e+190], N[(N[(q$95$m * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.89999999999999989e190

   1. Initial program 45.9%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 34.0

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

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 17.5%

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

   9. Taylor expanded in q around inf

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

   11. Applied rewrites 20.3%

       \[\leadsto \left(q \cdot 2\right) \cdot 0.5 \]

   if 2.89999999999999989e190 < r

   1. Initial program 8.4%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 17.2

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 17.2%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 13.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -0.46:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (if (<= p -0.46) (* -0.5 p) (* 0.5 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 <= -0.46) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (p <= (-0.46d0)) then
        tmp = (-0.5d0) * p
    else
        tmp = 0.5d0 * r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -0.46) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if p <= -0.46:
		tmp = -0.5 * p
	else:
		tmp = 0.5 * 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 <= -0.46)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (p <= -0.46)
		tmp = -0.5 * p;
	else
		tmp = 0.5 * r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -0.46], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -0.46000000000000002

   1. Initial program 38.3%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 13.8

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

   5. Applied rewrites 13.8%

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

   if -0.46000000000000002 < p

   1. Initial program 43.6%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 5.3

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 5.3%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 8.6% accurate, 41.7× speedup

Math

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

FPCore

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

C

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

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 = (-0.5d0) * p
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 -0.5 * p;
}

Python

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

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-0.5 * p)
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = -0.5 * p;
end

Wolfram

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

Derivation

1. Initial program 42.4%

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

3. Taylor expanded in p around -inf

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

   1. lower-*.f64 4.9

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

5. Applied rewrites 4.9%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
6. Add Preprocessing

Alternative 13: 1.2% accurate, 83.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 19.2

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

5. Applied rewrites 19.2%

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

Reproduce

herbie shell --seed 2024302 
(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)))))))