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

Percentage Accurate: 45.3% → 81.0%

Time: 6.9s

Alternatives: 9

Speedup: 14.7×

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: 45.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5.2 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 5.2e+166)
   (* 0.5 (+ (+ r (fabs p)) (- (fabs r) p)))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 5.2e+166) {
		tmp = 0.5 * ((r + fabs(p)) + (fabs(r) - p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 5.2e+166)
		tmp = Float64(0.5 * Float64(Float64(r + abs(p)) + Float64(abs(r) - p)));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 5.2e+166], N[(0.5 * N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 5.1999999999999999e166

   1. Initial program 50.3%

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

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

   5. Applied rewrites 33.9%

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

   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 38.6%

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

   if 5.1999999999999999e166 < q

   1. Initial program 7.6%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 84.8%

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

Alternative 2: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 4 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 4e-70)
   (* 0.5 (+ (+ r (fabs r)) (fabs p)))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 4e-70) {
		tmp = 0.5 * ((r + fabs(r)) + fabs(p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 4e-70)
		tmp = Float64(0.5 * Float64(Float64(r + abs(r)) + abs(p)));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 4e-70], N[(0.5 * N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 3.99999999999999998e-70

   1. Initial program 58.9%

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

   3. Taylor expanded in r around 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 42.1%

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

   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 45.9%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 32.4%

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

   if 3.99999999999999998e-70 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 34.7%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 31.2%

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

Alternative 3: 64.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -6.5 \cdot 10^{-64}:\\ \;\;\;\;\left(\left|r\right| - \left(p - \left|p\right|\right)\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -1.05 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|r\right| + r\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= p -6.5e-64)
   (* (- (fabs r) (- p (fabs p))) 0.5)
   (if (<= p -1.05e-299)
     (fma 0.5 (+ (fabs r) (fabs p)) q_m)
     (* 0.5 (+ (fabs r) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -6.5e-64) {
		tmp = (fabs(r) - (p - fabs(p))) * 0.5;
	} else if (p <= -1.05e-299) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = 0.5 * (fabs(r) + r);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (p <= -6.5e-64)
		tmp = Float64(Float64(abs(r) - Float64(p - abs(p))) * 0.5);
	elseif (p <= -1.05e-299)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(0.5 * Float64(abs(r) + r));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[p, -6.5e-64], N[(N[(N[Abs[r], $MachinePrecision] - N[(p - N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, -1.05e-299], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < -6.5000000000000004e-64

   1. Initial program 40.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 49.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 58.4%

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

   if -6.5000000000000004e-64 < p < -1.05000000000000005e-299

   1. Initial program 57.6%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 36.4%

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

   if -1.05000000000000005e-299 < p

   1. Initial program 45.3%

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

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

   5. Applied rewrites 24.9%

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

   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 24.5%

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

   10. Applied rewrites 24.5%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 24.9%

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

Alternative 4: 59.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|r\right| + r\right)\\ \end{array} \end{array} \]

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 2.25e+136)
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)
   (* 0.5 (+ (fabs r) 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 <= 2.25e+136) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = 0.5 * (fabs(r) + 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 <= 2.25e+136)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(0.5 * Float64(abs(r) + 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, 2.25e+136], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.25e136

   1. Initial program 51.5%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 25.4

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 27.6%

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

   if 2.25e136 < r

   1. Initial program 15.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 78.8%

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

   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 78.3%

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

   10. Applied rewrites 73.3%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 74.1%

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

Alternative 5: 55.8% accurate, 14.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right|, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|r\right| + r\right)\\ \end{array} \end{array} \]

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 1.5e+39) (fma (fabs r) 0.5 q_m) (* 0.5 (+ (fabs r) 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 <= 1.5e+39) {
		tmp = fma(fabs(r), 0.5, q_m);
	} else {
		tmp = 0.5 * (fabs(r) + 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 <= 1.5e+39)
		tmp = fma(abs(r), 0.5, q_m);
	else
		tmp = Float64(0.5 * Float64(abs(r) + 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, 1.5e+39], N[(N[Abs[r], $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.5e39

   1. Initial program 50.2%

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

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 25.9

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

   5. Applied rewrites 25.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 27.9%

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

   10. Applied rewrites 24.3%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 22.1%

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

   if 1.5e39 < r

   1. Initial program 32.6%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   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 70.5%

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

   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 70.1%

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

   10. Applied rewrites 58.2%

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

   11. Taylor expanded in p around 0

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

   13. Applied rewrites 59.2%

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

Alternative 6: 13.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 4.9 \cdot 10^{-55}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

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 (<= r 4.9e-55) (* -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 (r <= 4.9e-55) {
		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 (r <= 4.9d-55) 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 (r <= 4.9e-55) {
		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 r <= 4.9e-55:
		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 (r <= 4.9e-55)
		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 (r <= 4.9e-55)
		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[r, 4.9e-55], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.90000000000000035e-55

   1. Initial program 50.8%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 5.8

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

   5. Applied rewrites 5.8%

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

   if 4.90000000000000035e-55 < r

   1. Initial program 35.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 12.3

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

   5. Applied rewrites 12.3%

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

Alternative 7: 40.9% accurate, 27.8× speedup

Math

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

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 (fma (fabs 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) {
	return fma(fabs(r), 0.5, q_m);
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return fma(abs(r), 0.5, 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_] := N[(N[Abs[r], $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision]

Derivation

1. Initial program 45.8%

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

3. Taylor expanded in q around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-fabs.f64 23.8

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

5. Applied rewrites 23.8%

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

6. Taylor expanded in p around 0

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

8. Applied rewrites 26.4%

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

10. Applied rewrites 23.1%

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

11. Taylor expanded in p around 0

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

13. Applied rewrites 21.3%

    \[\leadsto \mathsf{fma}\left(\left|r\right|, 0.5, q\right) \]
14. Add Preprocessing

Alternative 8: 8.7% 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 45.8%

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

3. Taylor expanded in 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 45.8%

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

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 19.4

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

5. Applied rewrites 19.4%

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

Reproduce

herbie shell --seed 2024337 
(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)))))))