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

Percentage Accurate: 45.6% → 80.6%

Time: 7.0s

Alternatives: 9

Speedup: 11.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 45.6% 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: 80.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| - p, 0.5, r\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| + r, 0.5, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e+83)
   (fma (- (fabs p) p) 0.5 r)
   (fma (+ (fabs p) r) 0.5 q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e+83) {
		tmp = fma((fabs(p) - p), 0.5, r);
	} else {
		tmp = fma((fabs(p) + r), 0.5, 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) <= 1e+83)
		tmp = fma(Float64(abs(p) - p), 0.5, r);
	else
		tmp = fma(Float64(abs(p) + r), 0.5, 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], 1e+83], N[(N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision] * 0.5 + r), $MachinePrecision], N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000003e83

   1. Initial program 58.1%

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

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

   5. Applied rewrites 46.8%

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

   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 54.3%

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 54.0%

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

   if 1.00000000000000003e83 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 24.5%

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

   3. Taylor expanded in r around inf

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

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   7. 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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-/.f64 38.5

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

   8. Applied rewrites 38.5%

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

   10. Applied rewrites 36.5%

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

   11. Taylor expanded in q around 0

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

   13. Applied rewrites 36.5%

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

Alternative 2: 57.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|p\right|, r\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r + p, 0.5, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e-28) (fma 0.5 (fabs p) r) (fma (+ r p) 0.5 q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e-28) {
		tmp = fma(0.5, fabs(p), r);
	} else {
		tmp = fma((r + p), 0.5, 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) <= 1e-28)
		tmp = fma(0.5, abs(p), r);
	else
		tmp = fma(Float64(r + p), 0.5, 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], 1e-28], N[(0.5 * N[Abs[p], $MachinePrecision] + r), $MachinePrecision], N[(N[(r + p), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.99999999999999971e-29

   1. Initial program 55.1%

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

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

   5. Applied rewrites 48.3%

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

   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 56.7%

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

   10. Applied rewrites 56.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 36.2%

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

   if 9.99999999999999971e-29 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 32.1%

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

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

   5. Applied rewrites 14.7%

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

   6. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   7. 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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-/.f64 38.3

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

   8. Applied rewrites 38.3%

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

   9. Taylor expanded in q around 0

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

   11. Applied rewrites 38.3%

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

   13. Applied rewrites 35.2%

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

Alternative 3: 56.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|p\right|, r\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\_m\\ \end{array} \end{array} \]

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) 1e-28) (fma 0.5 (fabs p) r) (* 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) <= 1e-28) {
		tmp = fma(0.5, fabs(p), r);
	} else {
		tmp = 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) <= 1e-28)
		tmp = fma(0.5, abs(p), r);
	else
		tmp = Float64(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], 1e-28], N[(0.5 * N[Abs[p], $MachinePrecision] + r), $MachinePrecision], N[(1.0 * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.99999999999999971e-29

   1. Initial program 55.1%

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

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

   5. Applied rewrites 48.3%

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

   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 56.7%

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

   10. Applied rewrites 56.4%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 36.2%

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

   if 9.99999999999999971e-29 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 32.1%

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

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

   5. Applied rewrites 14.7%

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

   6. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   7. 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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-/.f64 38.3

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

   8. Applied rewrites 38.3%

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

   9. Taylor expanded in q around inf

      \[\leadsto 1 \cdot q \]
   10. Step-by-step derivation

   11. Applied rewrites 32.4%

       \[\leadsto 1 \cdot q \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 37.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\_m\\ \end{array} \end{array} \]

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) 5e-239) (* -0.5 p) (* 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) <= 5e-239) {
		tmp = -0.5 * p;
	} else {
		tmp = 1.0 * q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 5d-239) then
        tmp = (-0.5d0) * p
    else
        tmp = 1.0d0 * q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 5e-239) {
		tmp = -0.5 * p;
	} else {
		tmp = 1.0 * q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 5e-239:
		tmp = -0.5 * p
	else:
		tmp = 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) <= 5e-239)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(1.0 * q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 5e-239)
		tmp = -0.5 * p;
	else
		tmp = 1.0 * q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-239], N[(-0.5 * p), $MachinePrecision], N[(1.0 * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 5e-239

   1. Initial program 52.6%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 8.3

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

   5. Applied rewrites 8.3%

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

   if 5e-239 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 39.2%

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

   3. Taylor expanded in r around 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

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

   5. Applied rewrites 21.8%

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

   6. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   7. 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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in q around inf

      \[\leadsto 1 \cdot q \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto 1 \cdot q \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -6 \cdot 10^{-252}:\\ \;\;\;\;\left(\left|p\right| - p\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 3.5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| + r, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|p\right|, r\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r -6e-252)
   (* (- (fabs p) p) 0.5)
   (if (<= r 3.5e+83) (fma (+ (fabs p) r) 0.5 q_m) (fma 0.5 (fabs p) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= -6e-252) {
		tmp = (fabs(p) - p) * 0.5;
	} else if (r <= 3.5e+83) {
		tmp = fma((fabs(p) + r), 0.5, q_m);
	} else {
		tmp = fma(0.5, fabs(p), r);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= -6e-252)
		tmp = Float64(Float64(abs(p) - p) * 0.5);
	elseif (r <= 3.5e+83)
		tmp = fma(Float64(abs(p) + r), 0.5, q_m);
	else
		tmp = fma(0.5, abs(p), r);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, -6e-252], N[(N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 3.5e+83], N[(N[(N[Abs[p], $MachinePrecision] + r), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision], N[(0.5 * N[Abs[p], $MachinePrecision] + r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -5.9999999999999999e-252

   1. Initial program 44.0%

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

   3. Taylor expanded in r around inf

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

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

   5. Applied rewrites 19.0%

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

   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 24.4%

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

   10. Applied rewrites 23.7%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 24.8%

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

   if -5.9999999999999999e-252 < r < 3.49999999999999977e83

   1. Initial program 54.0%

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

   3. Taylor expanded in r around inf

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

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   7. 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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

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

   10. Applied rewrites 36.2%

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

   11. Taylor expanded in q around 0

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

   13. Applied rewrites 38.2%

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

   if 3.49999999999999977e83 < r

   1. Initial program 20.7%

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

   3. Taylor expanded in r around 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

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

   5. Applied rewrites 81.2%

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

   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 81.2%

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

   10. Applied rewrites 81.2%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 75.4%

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

Alternative 6: 64.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -5.5 \cdot 10^{-252}:\\ \;\;\;\;\left(\left|p\right| - p\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 2.15 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(r + p, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|p\right|, r\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r -5.5e-252)
   (* (- (fabs p) p) 0.5)
   (if (<= r 2.15e+46) (fma (+ r p) 0.5 q_m) (fma 0.5 (fabs p) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= -5.5e-252) {
		tmp = (fabs(p) - p) * 0.5;
	} else if (r <= 2.15e+46) {
		tmp = fma((r + p), 0.5, q_m);
	} else {
		tmp = fma(0.5, fabs(p), r);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= -5.5e-252)
		tmp = Float64(Float64(abs(p) - p) * 0.5);
	elseif (r <= 2.15e+46)
		tmp = fma(Float64(r + p), 0.5, q_m);
	else
		tmp = fma(0.5, abs(p), r);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, -5.5e-252], N[(N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 2.15e+46], N[(N[(r + p), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision], N[(0.5 * N[Abs[p], $MachinePrecision] + r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -5.5e-252

   1. Initial program 44.0%

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

   3. Taylor expanded in r around inf

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

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

   5. Applied rewrites 19.0%

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

   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 24.4%

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

   10. Applied rewrites 23.7%

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

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 24.8%

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

   if -5.5e-252 < r < 2.15000000000000002e46

   1. Initial program 53.5%

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

   3. Taylor expanded in r around inf

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

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

   5. Applied rewrites 23.7%

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

   6. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   7. 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. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-/.f64 35.5

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in q around 0

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

   11. Applied rewrites 37.5%

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

   13. Applied rewrites 34.8%

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

   if 2.15000000000000002e46 < r

   1. Initial program 25.1%

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

   3. Taylor expanded in 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

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

   5. Applied rewrites 79.2%

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

   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 79.2%

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

   10. Applied rewrites 79.2%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 70.6%

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

Alternative 7: 13.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

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 -1.7e-10) (* -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 <= -1.7e-10) {
		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 <= (-1.7d-10)) 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 <= -1.7e-10) {
		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 <= -1.7e-10:
		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 <= -1.7e-10)
		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 <= -1.7e-10)
		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, -1.7e-10], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -1.70000000000000007e-10

   1. Initial program 40.1%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 13.1

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

   5. Applied rewrites 13.1%

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

   if -1.70000000000000007e-10 < p

   1. Initial program 45.5%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 5.7

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

   5. Applied rewrites 5.7%

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

Alternative 8: 8.5% 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 44.0%

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

3. Taylor expanded in p around -inf

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

   1. lower-*.f64 5.3

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

5. Applied rewrites 5.3%

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

Alternative 9: 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 44.0%

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

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 18.8

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

5. Applied rewrites 18.8%

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

Reproduce

herbie shell --seed 2024333 
(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)))))))