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

Percentage Accurate: 24.4% → 61.6%

Time: 9.9s

Alternatives: 9

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: 24.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -4.9 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{p}{r} + \frac{\left|p\right|}{r}\right) + \frac{\left|r\right|}{r}, r, -r\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;r \leq 1.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.5, \frac{\left|p\right| + \left|r\right|}{q\_m}, -1\right)} \cdot q\_m\\ \mathbf{elif}\;r \leq 1.05 \cdot 10^{+273}:\\ \;\;\;\;\left(\left(\left(\left|r\right| - r\right) + p\right) + \left|p\right|\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{q\_m}{r} \cdot q\_m\right) \cdot -2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r -4.9e-173)
   (* (fma (+ (+ (/ p r) (/ (fabs p) r)) (/ (fabs r) r)) r (- r)) (/ 1.0 2.0))
   (if (<= r 1.4e+189)
     (* (/ 1.0 (fma -0.5 (/ (+ (fabs p) (fabs r)) q_m) -1.0)) q_m)
     (if (<= r 1.05e+273)
       (* (+ (+ (- (fabs r) r) p) (fabs p)) (/ 1.0 2.0))
       (* (* (* (/ q_m r) q_m) -2.0) 0.5)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= -4.9e-173) {
		tmp = fma((((p / r) + (fabs(p) / r)) + (fabs(r) / r)), r, -r) * (1.0 / 2.0);
	} else if (r <= 1.4e+189) {
		tmp = (1.0 / fma(-0.5, ((fabs(p) + fabs(r)) / q_m), -1.0)) * q_m;
	} else if (r <= 1.05e+273) {
		tmp = (((fabs(r) - r) + p) + fabs(p)) * (1.0 / 2.0);
	} else {
		tmp = (((q_m / r) * q_m) * -2.0) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= -4.9e-173)
		tmp = Float64(fma(Float64(Float64(Float64(p / r) + Float64(abs(p) / r)) + Float64(abs(r) / r)), r, Float64(-r)) * Float64(1.0 / 2.0));
	elseif (r <= 1.4e+189)
		tmp = Float64(Float64(1.0 / fma(-0.5, Float64(Float64(abs(p) + abs(r)) / q_m), -1.0)) * q_m);
	elseif (r <= 1.05e+273)
		tmp = Float64(Float64(Float64(Float64(abs(r) - r) + p) + abs(p)) * Float64(1.0 / 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(q_m / r) * q_m) * -2.0) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, -4.9e-173], N[(N[(N[(N[(N[(p / r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] * r + (-r)), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.4e+189], N[(N[(1.0 / N[(-0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision], If[LessEqual[r, 1.05e+273], N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(q$95$m / r), $MachinePrecision] * q$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if r < -4.89999999999999991e-173

   1. Initial program 25.6%

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

   3. Taylor expanded in r around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 17.1

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

   5. Applied rewrites 17.1%

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

   6. Taylor expanded in r around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 6.1%

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

   10. Applied rewrites 19.6%

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

   if -4.89999999999999991e-173 < r < 1.40000000000000003e189

   1. Initial program 30.7%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 27.4%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left|p\right| + \left|r\right|, \frac{0.5}{q}, 1\right)}{\mathsf{fma}\left({\left(\frac{q}{\left|p\right| + \left|r\right|}\right)}^{-2}, 0.25, -1\right)}} \cdot q \]

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 45.5%

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

   if 1.40000000000000003e189 < r < 1.05000000000000001e273

   1. Initial program 2.0%

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

   3. Taylor expanded in r around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.6

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in r around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 19.6%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 67.6%

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

   if 1.05000000000000001e273 < r

   1. Initial program 2.7%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. associate--r+ N/A

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

      7. associate-+r+ N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 76.1%

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

   10. Applied rewrites 99.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 99.4

          \[\leadsto \color{blue}{\left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \cdot 0.5} \]

   12. Applied rewrites 99.4%

       \[\leadsto \color{blue}{\left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -4.9 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{p}{r} + \frac{\left|p\right|}{r}\right) + \frac{\left|r\right|}{r}, r, -r\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;r \leq 1.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.5, \frac{\left|p\right| + \left|r\right|}{q}, -1\right)} \cdot q\\ \mathbf{elif}\;r \leq 1.05 \cdot 10^{+273}:\\ \;\;\;\;\left(\left(\left(\left|r\right| - r\right) + p\right) + \left|p\right|\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| - r\\ \mathbf{if}\;{q\_m}^{2} \leq 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0 + \left|p\right|}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{elif}\;{q\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 + p\right) + \left|p\right|, r, \left(q\_m \cdot q\_m\right) \cdot -2\right)}{r} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.5, \frac{\left|p\right| + \left|r\right|}{q\_m}, -1\right)} \cdot q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (- (fabs r) r)))
   (if (<= (pow q_m 2.0) 1e-220)
     (* (fma (/ (+ t_0 (fabs p)) p) -0.5 -0.5) (- p))
     (if (<= (pow q_m 2.0) 2e+94)
       (*
        (/ (fma (+ (+ t_0 p) (fabs p)) r (* (* q_m q_m) -2.0)) r)
        (/ 1.0 2.0))
       (* (/ 1.0 (fma -0.5 (/ (+ (fabs p) (fabs r)) q_m) -1.0)) q_m)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) - r;
	double tmp;
	if (pow(q_m, 2.0) <= 1e-220) {
		tmp = fma(((t_0 + fabs(p)) / p), -0.5, -0.5) * -p;
	} else if (pow(q_m, 2.0) <= 2e+94) {
		tmp = (fma(((t_0 + p) + fabs(p)), r, ((q_m * q_m) * -2.0)) / r) * (1.0 / 2.0);
	} else {
		tmp = (1.0 / fma(-0.5, ((fabs(p) + fabs(r)) / q_m), -1.0)) * q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(r) - r)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 1e-220)
		tmp = Float64(fma(Float64(Float64(t_0 + abs(p)) / p), -0.5, -0.5) * Float64(-p));
	elseif ((q_m ^ 2.0) <= 2e+94)
		tmp = Float64(Float64(fma(Float64(Float64(t_0 + p) + abs(p)), r, Float64(Float64(q_m * q_m) * -2.0)) / r) * Float64(1.0 / 2.0));
	else
		tmp = Float64(Float64(1.0 / fma(-0.5, Float64(Float64(abs(p) + abs(r)) / q_m), -1.0)) * q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]}, If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-220], N[(N[(N[(N[(t$95$0 + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5 + -0.5), $MachinePrecision] * (-p)), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[(N[(N[(N[(t$95$0 + p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * r + N[(N[(q$95$m * q$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(-0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.99999999999999992e-221

   1. Initial program 26.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 8.1

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

   5. Applied rewrites 8.1%

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

   7. Applied rewrites 6.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left|p\right| + \left|r\right|, \frac{0.5}{q}, 1\right)}{\mathsf{fma}\left({\left(\frac{q}{\left|p\right| + \left|r\right|}\right)}^{-2}, 0.25, -1\right)}} \cdot q \]

   8. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 36.8%

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

   if 9.99999999999999992e-221 < (pow.f64 q #s(literal 2 binary64)) < 2e94

   1. Initial program 23.9%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. associate--r+ N/A

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

      7. associate-+r+ N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 15.0%

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

   10. Applied rewrites 15.0%

       \[\leadsto \frac{1}{2} \cdot \left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \]

   11. Taylor expanded in r around 0

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

   13. Applied rewrites 18.4%

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

   if 2e94 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.3%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 35.4

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

   5. Applied rewrites 35.4%

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

   7. Applied rewrites 35.4%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left|p\right| + \left|r\right|, \frac{0.5}{q}, 1\right)}{\mathsf{fma}\left({\left(\frac{q}{\left|p\right| + \left|r\right|}\right)}^{-2}, 0.25, -1\right)}} \cdot q \]

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 40.0%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{elif}\;{q}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\left|r\right| - r\right) + p\right) + \left|p\right|, r, \left(q \cdot q\right) \cdot -2\right)}{r} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.5, \frac{\left|p\right| + \left|r\right|}{q}, -1\right)} \cdot q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.5, \frac{\left|p\right| + \left|r\right|}{q\_m}, -1\right)} \cdot q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e-187)
   (* (fma (/ (+ (- (fabs r) r) (fabs p)) p) -0.5 -0.5) (- p))
   (* (/ 1.0 (fma -0.5 (/ (+ (fabs p) (fabs r)) q_m) -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-187) {
		tmp = fma((((fabs(r) - r) + fabs(p)) / p), -0.5, -0.5) * -p;
	} else {
		tmp = (1.0 / fma(-0.5, ((fabs(p) + fabs(r)) / q_m), -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-187)
		tmp = Float64(fma(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p), -0.5, -0.5) * Float64(-p));
	else
		tmp = Float64(Float64(1.0 / fma(-0.5, Float64(Float64(abs(p) + abs(r)) / q_m), -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-187], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5 + -0.5), $MachinePrecision] * (-p)), $MachinePrecision], N[(N[(1.0 / N[(-0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] / q$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1e-187

   1. Initial program 25.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 8.5

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

   5. Applied rewrites 8.5%

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

   7. Applied rewrites 6.7%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left|p\right| + \left|r\right|, \frac{0.5}{q}, 1\right)}{\mathsf{fma}\left({\left(\frac{q}{\left|p\right| + \left|r\right|}\right)}^{-2}, 0.25, -1\right)}} \cdot q \]

   8. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 33.8%

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

   if 1e-187 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 28.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left|p\right| + \left|r\right|, \frac{0.5}{q}, 1\right)}{\mathsf{fma}\left({\left(\frac{q}{\left|p\right| + \left|r\right|}\right)}^{-2}, 0.25, -1\right)}} \cdot q \]

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 35.0%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.5, \frac{\left|p\right| + \left|r\right|}{q}, -1\right)} \cdot q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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
 (if (<= (pow q_m 2.0) 5e-144)
   (* (fma (/ (+ (- (fabs r) r) (fabs p)) p) -0.5 -0.5) (- p))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 5e-144) {
		tmp = fma((((fabs(r) - r) + fabs(p)) / p), -0.5, -0.5) * -p;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 5e-144)
		tmp = Float64(fma(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p), -0.5, -0.5) * Float64(-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_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-144], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5 + -0.5), $MachinePrecision] * (-p)), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.9999999999999998e-144

   1. Initial program 24.7%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 8.1

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

   5. Applied rewrites 8.1%

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

   7. Applied rewrites 6.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\left|p\right| + \left|r\right|, \frac{0.5}{q}, 1\right)}{\mathsf{fma}\left({\left(\frac{q}{\left|p\right| + \left|r\right|}\right)}^{-2}, 0.25, -1\right)}} \cdot q \]

   8. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 31.8%

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

   if 4.9999999999999998e-144 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.1%

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

   3. Taylor expanded in 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 31.0

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

   5. Applied rewrites 31.0%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\left(\left(\left(\left|r\right| - r\right) + p\right) + \left|p\right|\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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-144)
   (* (+ (+ (- (fabs r) r) p) (fabs p)) (/ 1.0 2.0))
   (- 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-144) {
		tmp = (((fabs(r) - r) + p) + fabs(p)) * (1.0 / 2.0);
	} 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-144) then
        tmp = (((abs(r) - r) + p) + abs(p)) * (1.0d0 / 2.0d0)
    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-144) {
		tmp = (((Math.abs(r) - r) + p) + Math.abs(p)) * (1.0 / 2.0);
	} 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-144:
		tmp = (((math.fabs(r) - r) + p) + math.fabs(p)) * (1.0 / 2.0)
	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-144)
		tmp = Float64(Float64(Float64(Float64(abs(r) - r) + p) + abs(p)) * Float64(1.0 / 2.0));
	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-144)
		tmp = (((abs(r) - r) + p) + abs(p)) * (1.0 / 2.0);
	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-144], N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.9999999999999998e-144

   1. Initial program 24.7%

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

   3. Taylor expanded in r around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-+r+ N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 15.7

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in r around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

   8. Applied rewrites 16.2%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 31.8%

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

   if 4.9999999999999998e-144 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.1%

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

   3. Taylor expanded in 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 31.0

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

   5. Applied rewrites 31.0%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\left(\left(\left(\left|r\right| - r\right) + p\right) + \left|p\right|\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{q\_m}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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) 0.0) (/ (* (- q_m) q_m) q_m) (- 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) <= 0.0) {
		tmp = (-q_m * q_m) / q_m;
	} 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) <= 0.0d0) then
        tmp = (-q_m * q_m) / q_m
    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) <= 0.0) {
		tmp = (-q_m * q_m) / q_m;
	} 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) <= 0.0:
		tmp = (-q_m * q_m) / q_m
	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) <= 0.0)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / q_m);
	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) <= 0.0)
		tmp = (-q_m * q_m) / q_m;
	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], 0.0], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / q$95$m), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 0.0

   1. Initial program 25.6%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 8.5

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

   5. Applied rewrites 8.5%

      \[\leadsto \color{blue}{-q} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.4%

      \[\leadsto \frac{\left(-q\right) \cdot q}{\color{blue}{0 + q}} \]

   if 0.0 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 26.2%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.4

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

   5. Applied rewrites 26.4%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 0:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{q}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.1 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{q\_m}\\ \mathbf{elif}\;q\_m \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;\left(\left(\frac{q\_m}{r} \cdot q\_m\right) \cdot -2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= q_m 1.1e-159)
   (/ (* (- q_m) q_m) q_m)
   (if (<= q_m 2.6e+52) (* (* (* (/ q_m r) q_m) -2.0) 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 (q_m <= 1.1e-159) {
		tmp = (-q_m * q_m) / q_m;
	} else if (q_m <= 2.6e+52) {
		tmp = (((q_m / r) * q_m) * -2.0) * 0.5;
	} 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 <= 1.1d-159) then
        tmp = (-q_m * q_m) / q_m
    else if (q_m <= 2.6d+52) then
        tmp = (((q_m / r) * q_m) * (-2.0d0)) * 0.5d0
    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 (q_m <= 1.1e-159) {
		tmp = (-q_m * q_m) / q_m;
	} else if (q_m <= 2.6e+52) {
		tmp = (((q_m / r) * q_m) * -2.0) * 0.5;
	} 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 q_m <= 1.1e-159:
		tmp = (-q_m * q_m) / q_m
	elif q_m <= 2.6e+52:
		tmp = (((q_m / r) * q_m) * -2.0) * 0.5
	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 <= 1.1e-159)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / q_m);
	elseif (q_m <= 2.6e+52)
		tmp = Float64(Float64(Float64(Float64(q_m / r) * q_m) * -2.0) * 0.5);
	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 <= 1.1e-159)
		tmp = (-q_m * q_m) / q_m;
	elseif (q_m <= 2.6e+52)
		tmp = (((q_m / r) * q_m) * -2.0) * 0.5;
	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[q$95$m, 1.1e-159], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / q$95$m), $MachinePrecision], If[LessEqual[q$95$m, 2.6e+52], N[(N[(N[(N[(q$95$m / r), $MachinePrecision] * q$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if q < 1.1e-159

   1. Initial program 25.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 5.6

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

   5. Applied rewrites 5.6%

      \[\leadsto \color{blue}{-q} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.3%

      \[\leadsto \frac{\left(-q\right) \cdot q}{\color{blue}{0 + q}} \]

   if 1.1e-159 < q < 2.6e52

   1. Initial program 27.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. associate--r+ N/A

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

      7. associate-+r+ N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 19.3%

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

   10. Applied rewrites 19.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 19.5

          \[\leadsto \color{blue}{\left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \cdot 0.5} \]

   12. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \cdot 0.5} \]

   if 2.6e52 < q

   1. Initial program 26.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.0

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

   5. Applied rewrites 71.0%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 1.1 \cdot 10^{-159}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{q}\\ \mathbf{elif}\;q \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;\left(\left(\frac{q}{r} \cdot q\right) \cdot -2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{q\_m \cdot q\_m}{r} \cdot -2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= q_m 2.6e+52) (* (* (/ (* q_m q_m) r) -2.0) 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 (q_m <= 2.6e+52) {
		tmp = (((q_m * q_m) / r) * -2.0) * 0.5;
	} 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 <= 2.6d+52) then
        tmp = (((q_m * q_m) / r) * (-2.0d0)) * 0.5d0
    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 (q_m <= 2.6e+52) {
		tmp = (((q_m * q_m) / r) * -2.0) * 0.5;
	} 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 q_m <= 2.6e+52:
		tmp = (((q_m * q_m) / r) * -2.0) * 0.5
	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 <= 2.6e+52)
		tmp = Float64(Float64(Float64(Float64(q_m * q_m) / r) * -2.0) * 0.5);
	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 <= 2.6e+52)
		tmp = (((q_m * q_m) / r) * -2.0) * 0.5;
	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[q$95$m, 2.6e+52], N[(N[(N[(N[(q$95$m * q$95$m), $MachinePrecision] / r), $MachinePrecision] * -2.0), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 2.6e52

   1. Initial program 25.8%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. associate--r+ N/A

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

      7. associate-+r+ N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 25.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.5

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

      4. lift-/.f64 N/A

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

      5. metadata-eval 25.5

         \[\leadsto \left(\frac{q \cdot q}{r} \cdot -2\right) \cdot \color{blue}{0.5} \]

   10. Applied rewrites 25.5%

       \[\leadsto \color{blue}{\left(\frac{q \cdot q}{r} \cdot -2\right) \cdot 0.5} \]

   if 2.6e52 < q

   1. Initial program 26.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.0

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

   5. Applied rewrites 71.0%

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

Alternative 9: 34.7% 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 26.0%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 21.3

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

5. Applied rewrites 21.3%

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

Reproduce

herbie shell --seed 2024304 
(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)))))))