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

Percentage Accurate: 24.0% → 57.4%

Time: 8.0s

Alternatives: 5

Speedup: 83.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.0% 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: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, 0.5, 0.5\right)\\ \mathbf{if}\;{q\_m}^{2} \leq 100:\\ \;\;\;\;t\_0 \cdot p\\ \mathbf{elif}\;{q\_m}^{2} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\left(q\_m \cdot q\_m\right) \cdot \mathsf{fma}\left(\frac{t\_0}{q\_m}, \frac{p}{q\_m}, \frac{1}{p}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (fma (/ (+ (fabs p) (- (fabs r) r)) p) 0.5 0.5)))
   (if (<= (pow q_m 2.0) 100.0)
     (* t_0 p)
     (if (<= (pow q_m 2.0) 5e+48)
       (* (* q_m q_m) (fma (/ t_0 q_m) (/ p q_m) (/ 1.0 p)))
       (- q_m)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fma(((fabs(p) + (fabs(r) - r)) / p), 0.5, 0.5);
	double tmp;
	if (pow(q_m, 2.0) <= 100.0) {
		tmp = t_0 * p;
	} else if (pow(q_m, 2.0) <= 5e+48) {
		tmp = (q_m * q_m) * fma((t_0 / q_m), (p / q_m), (1.0 / p));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = fma(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p), 0.5, 0.5)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 100.0)
		tmp = Float64(t_0 * p);
	elseif ((q_m ^ 2.0) <= 5e+48)
		tmp = Float64(Float64(q_m * q_m) * fma(Float64(t_0 / q_m), Float64(p / q_m), Float64(1.0 / p)));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 100.0], N[(t$95$0 * p), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e+48], N[(N[(q$95$m * q$95$m), $MachinePrecision] * N[(N[(t$95$0 / q$95$m), $MachinePrecision] * N[(p / q$95$m), $MachinePrecision] + N[(1.0 / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 100

   1. Initial program 24.0%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 12.4%

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

   6. Taylor expanded in p around inf

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

   8. Applied rewrites 30.0%

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

   if 100 < (pow.f64 q #s(literal 2 binary64)) < 4.99999999999999973e48

   1. Initial program 12.1%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in q around inf

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

   8. Applied rewrites 21.9%

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

   if 4.99999999999999973e48 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.8%

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

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

   5. Applied rewrites 32.1%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, 0.5, 0.5\right) \cdot p\\ \mathbf{elif}\;{q}^{2} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\left(q \cdot q\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, 0.5, 0.5\right)}{q}, \frac{p}{q}, \frac{1}{p}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, 0.5, 0.5\right) \cdot p\\ \mathbf{elif}\;{q\_m}^{2} \leq 8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 2e-58)
   (* (fma (/ (+ (fabs p) (- (fabs r) r)) p) 0.5 0.5) p)
   (if (<= (pow q_m 2.0) 8e+31) (/ (* (- q_m) q_m) r) (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 2e-58) {
		tmp = fma(((fabs(p) + (fabs(r) - r)) / p), 0.5, 0.5) * p;
	} else if (pow(q_m, 2.0) <= 8e+31) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 2e-58)
		tmp = Float64(fma(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p), 0.5, 0.5) * p);
	elseif ((q_m ^ 2.0) <= 8e+31)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e-58], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * p), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 8e+31], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 24.2%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 13.2%

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

   6. Taylor expanded in p around inf

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

   8. Applied rewrites 31.4%

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

   if 2.0000000000000001e-58 < (pow.f64 q #s(literal 2 binary64)) < 7.9999999999999997e31

   1. Initial program 10.8%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 7.5%

       \[\leadsto \frac{q \cdot q}{-r} \]

   if 7.9999999999999997e31 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 32.0

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

   5. Applied rewrites 32.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, 0.5, 0.5\right) \cdot p\\ \mathbf{elif}\;{q}^{2} \leq 8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 8e+31) (/ (* (- q_m) q_m) r) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 8e+31) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 8d+31) then
        tmp = (-q_m * q_m) / r
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 8e+31) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 8e+31:
		tmp = (-q_m * q_m) / r
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 8e+31)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 8e+31)
		tmp = (-q_m * q_m) / r;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 8e+31], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 7.9999999999999997e31

   1. Initial program 21.9%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 24.7%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 31.4%

       \[\leadsto \frac{q \cdot q}{-r} \]

   if 7.9999999999999997e31 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 32.0

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

   5. Applied rewrites 32.0%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\frac{q\_m \cdot q\_m}{p}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 5e+48) (/ (* q_m q_m) 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) <= 5e+48) {
		tmp = (q_m * q_m) / p;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 5d+48) then
        tmp = (q_m * q_m) / p
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 5e+48) {
		tmp = (q_m * q_m) / p;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 5e+48:
		tmp = (q_m * q_m) / p
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 5e+48)
		tmp = Float64(Float64(q_m * q_m) / p);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 5e+48)
		tmp = (q_m * q_m) / p;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e+48], N[(N[(q$95$m * q$95$m), $MachinePrecision] / p), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.99999999999999973e48

   1. Initial program 22.3%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in p around 0

      \[\leadsto \frac{{q}^{2}}{\color{blue}{p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.2%

      \[\leadsto \frac{q \cdot q}{\color{blue}{p}} \]

   if 4.99999999999999973e48 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.8%

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

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

   5. Applied rewrites 32.1%

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

Alternative 5: 36.2% 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 24.3%

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

3. Taylor expanded in 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 19.5

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

5. Applied rewrites 19.5%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))