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

Percentage Accurate: 24.3% → 70.3%

Time: 9.6s

Alternatives: 6

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: 24.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-186}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{elif}\;{q\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{-2 \cdot \left(q\_m \cdot q\_m\right)}{\sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)} + \left(\left|p\right| + \left|r\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e-186)
   (* 0.5 (+ (+ p (fabs p)) (- (fabs r) r)))
   (if (<= (pow q_m 2.0) 5e+275)
     (/
      (* -2.0 (* q_m q_m))
      (+ (sqrt (fma (* q_m q_m) 4.0 (* p p))) (+ (fabs p) (fabs r))))
     (* (- (+ (fabs r) (fabs p)) (* q_m 2.0)) 0.5))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e-186) {
		tmp = 0.5 * ((p + fabs(p)) + (fabs(r) - r));
	} else if (pow(q_m, 2.0) <= 5e+275) {
		tmp = (-2.0 * (q_m * q_m)) / (sqrt(fma((q_m * q_m), 4.0, (p * p))) + (fabs(p) + fabs(r)));
	} else {
		tmp = ((fabs(r) + fabs(p)) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 1e-186)
		tmp = Float64(0.5 * Float64(Float64(p + abs(p)) + Float64(abs(r) - r)));
	elseif ((q_m ^ 2.0) <= 5e+275)
		tmp = Float64(Float64(-2.0 * Float64(q_m * q_m)) / Float64(sqrt(fma(Float64(q_m * q_m), 4.0, Float64(p * p))) + Float64(abs(p) + abs(r))));
	else
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-186], N[(0.5 * N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e+275], N[(N[(-2.0 * N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999991e-187

   1. Initial program 20.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 7.7

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

   5. Applied rewrites 7.7%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 33.5

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 45.3%

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

   if 9.9999999999999991e-187 < (pow.f64 q #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 29.6%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 26.4

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

   5. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(\left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 26.0%

      \[\leadsto \frac{\left({\left(\left|p\right| + \left|r\right|\right)}^{2} - \mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)\right) \cdot 0.5}{\color{blue}{\sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)} + \left(\left|p\right| + \left|r\right|\right)}} \]

   8. Taylor expanded in q around inf

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

   10. Applied rewrites 41.7%

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

   if 5.0000000000000003e275 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 11.7

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 45.5%

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

Alternative 2: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{+136}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - q\_m \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e+136)
   (* 0.5 (+ (+ p (fabs p)) (- (fabs r) r)))
   (* (- (+ (fabs r) (fabs p)) (* q_m 2.0)) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e+136) {
		tmp = 0.5 * ((p + fabs(p)) + (fabs(r) - r));
	} else {
		tmp = ((fabs(r) + fabs(p)) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 1d+136) then
        tmp = 0.5d0 * ((p + abs(p)) + (abs(r) - r))
    else
        tmp = ((abs(r) + abs(p)) - (q_m * 2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 1e+136) {
		tmp = 0.5 * ((p + Math.abs(p)) + (Math.abs(r) - r));
	} else {
		tmp = ((Math.abs(r) + Math.abs(p)) - (q_m * 2.0)) * 0.5;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 1e+136:
		tmp = 0.5 * ((p + math.fabs(p)) + (math.fabs(r) - r))
	else:
		tmp = ((math.fabs(r) + math.fabs(p)) - (q_m * 2.0)) * 0.5
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 1e+136)
		tmp = Float64(0.5 * Float64(Float64(p + abs(p)) + Float64(abs(r) - r)));
	else
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - Float64(q_m * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 1e+136)
		tmp = 0.5 * ((p + abs(p)) + (abs(r) - r));
	else
		tmp = ((abs(r) + abs(p)) - (q_m * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+136], N[(0.5 * N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000006e136

   1. Initial program 23.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 11.1

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 25.0

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

   8. Applied rewrites 25.0%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 34.3%

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

   if 1.00000000000000006e136 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 18.8%

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

   3. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 17.2

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 37.6%

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

Alternative 3: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{+136}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \left|p\right|\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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) 1e+136)
   (* 0.5 (+ (+ p (fabs p)) (- (fabs r) r)))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e+136) {
		tmp = 0.5 * ((p + fabs(p)) + (fabs(r) - r));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 1d+136) then
        tmp = 0.5d0 * ((p + abs(p)) + (abs(r) - r))
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 1e+136) {
		tmp = 0.5 * ((p + Math.abs(p)) + (Math.abs(r) - r));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 1e+136:
		tmp = 0.5 * ((p + math.fabs(p)) + (math.fabs(r) - r))
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 1e+136)
		tmp = Float64(0.5 * Float64(Float64(p + abs(p)) + Float64(abs(r) - r)));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 1e+136)
		tmp = 0.5 * ((p + abs(p)) + (abs(r) - r));
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+136], N[(0.5 * N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000006e136

   1. Initial program 23.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 11.1

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 25.0

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

   8. Applied rewrites 25.0%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 34.3%

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

   if 1.00000000000000006e136 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 18.8%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.9

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

   5. Applied rewrites 37.9%

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

Alternative 4: 40.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-209}:\\ \;\;\;\;\left(\left|r\right| + \left(\left|p\right| - r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 5e-209) (* (+ (fabs r) (- (fabs p) r)) 0.5) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 5e-209) {
		tmp = (fabs(r) + (fabs(p) - r)) * 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 ** 2.0d0) <= 5d-209) then
        tmp = (abs(r) + (abs(p) - r)) * 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 (Math.pow(q_m, 2.0) <= 5e-209) {
		tmp = (Math.abs(r) + (Math.abs(p) - r)) * 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 math.pow(q_m, 2.0) <= 5e-209:
		tmp = (math.fabs(r) + (math.fabs(p) - r)) * 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 ^ 2.0) <= 5e-209)
		tmp = Float64(Float64(abs(r) + Float64(abs(p) - r)) * 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 ^ 2.0) <= 5e-209)
		tmp = (abs(r) + (abs(p) - r)) * 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[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-209], N[(N[(N[Abs[r], $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 5.0000000000000005e-209

   1. Initial program 20.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 6.9

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

   5. Applied rewrites 6.9%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 35.0

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

   8. Applied rewrites 35.0%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 13.1%

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

   if 5.0000000000000005e-209 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 22.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 27.3

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

   5. Applied rewrites 27.3%

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

Alternative 5: 40.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 2.7 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(p + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 2.7e-98) (* (+ (+ p (fabs r)) (fabs p)) 0.5) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 2.7e-98) {
		tmp = ((p + fabs(r)) + fabs(p)) * 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 <= 2.7d-98) then
        tmp = ((p + abs(r)) + abs(p)) * 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 <= 2.7e-98) {
		tmp = ((p + Math.abs(r)) + Math.abs(p)) * 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 <= 2.7e-98:
		tmp = ((p + math.fabs(r)) + math.fabs(p)) * 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 <= 2.7e-98)
		tmp = Float64(Float64(Float64(p + abs(r)) + abs(p)) * 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 <= 2.7e-98)
		tmp = ((p + abs(r)) + abs(p)) * 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, 2.7e-98], N[(N[(N[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 2.6999999999999999e-98

   1. Initial program 20.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 4.5

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 21.5

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 29.7%

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

   12. Taylor expanded in r around 0

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

   14. Applied rewrites 8.5%

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

   if 2.6999999999999999e-98 < q

   1. Initial program 24.5%

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

   3. Taylor expanded in 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 55.5

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

   5. Applied rewrites 55.5%

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

Alternative 6: 35.4% 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 21.6%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 21.1

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

5. Applied rewrites 21.1%

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

Reproduce

herbie shell --seed 2024313 
(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)))))))