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

Percentage Accurate: 24.3% → 62.6%

Time: 10.2s

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: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{+110}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(2, p \cdot \frac{r}{\left|r\right| + r}, \left|p\right|\right) + -4 \cdot \frac{q\_m \cdot q\_m}{\sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, r \cdot r\right)} + \left|r\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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+110)
   (*
    (pow 2.0 -1.0)
    (+
     (fma 2.0 (* p (/ r (+ (fabs r) r))) (fabs p))
     (*
      -4.0
      (/ (* q_m q_m) (+ (sqrt (fma (* q_m q_m) 4.0 (* r r))) (fabs 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+110) {
		tmp = pow(2.0, -1.0) * (fma(2.0, (p * (r / (fabs(r) + r))), fabs(p)) + (-4.0 * ((q_m * q_m) / (sqrt(fma((q_m * q_m), 4.0, (r * r))) + fabs(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+110)
		tmp = Float64((2.0 ^ -1.0) * Float64(fma(2.0, Float64(p * Float64(r / Float64(abs(r) + r))), abs(p)) + Float64(-4.0 * Float64(Float64(q_m * q_m) / Float64(sqrt(fma(Float64(q_m * q_m), 4.0, Float64(r * r))) + abs(r))))));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+110], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(2.0 * N[(p * N[(r / N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(q$95$m * q$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(r * r), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1e110

   1. Initial program 24.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-- N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in p around 0

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in q around 0

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

   10. Applied rewrites 34.8%

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

   if 1e110 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 23.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 34.8

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

   5. Applied rewrites 34.8%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{+110}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(2, p \cdot \frac{r}{\left|r\right| + r}, \left|p\right|\right) + -4 \cdot \frac{q \cdot q}{\sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)} + \left|r\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{+114}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(2, p \cdot \frac{r}{\left|r\right| + r}, \left|p\right|\right) + -4 \cdot \frac{q\_m \cdot q\_m}{\mathsf{fma}\left(\frac{q\_m \cdot q\_m}{r}, 2, r\right) + \left|r\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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+114)
   (*
    (pow 2.0 -1.0)
    (+
     (fma 2.0 (* p (/ r (+ (fabs r) r))) (fabs p))
     (* -4.0 (/ (* q_m q_m) (+ (fma (/ (* q_m q_m) r) 2.0 r) (fabs 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+114) {
		tmp = pow(2.0, -1.0) * (fma(2.0, (p * (r / (fabs(r) + r))), fabs(p)) + (-4.0 * ((q_m * q_m) / (fma(((q_m * q_m) / r), 2.0, r) + fabs(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+114)
		tmp = Float64((2.0 ^ -1.0) * Float64(fma(2.0, Float64(p * Float64(r / Float64(abs(r) + r))), abs(p)) + Float64(-4.0 * Float64(Float64(q_m * q_m) / Float64(fma(Float64(Float64(q_m * q_m) / r), 2.0, r) + abs(r))))));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+114], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(2.0 * N[(p * N[(r / N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(q$95$m * q$95$m), $MachinePrecision] / N[(N[(N[(N[(q$95$m * q$95$m), $MachinePrecision] / r), $MachinePrecision] * 2.0 + r), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1e114

   1. Initial program 24.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-- N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in p around 0

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in q around 0

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

   10. Applied rewrites 34.6%

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

   11. Taylor expanded in q around 0

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

   13. Applied rewrites 34.5%

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

   if 1e114 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 23.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 35.1

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

   5. Applied rewrites 35.1%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{+114}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(2, p \cdot \frac{r}{\left|r\right| + r}, \left|p\right|\right) + -4 \cdot \frac{q \cdot q}{\mathsf{fma}\left(\frac{q \cdot q}{r}, 2, r\right) + \left|r\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + r\\ \mathbf{if}\;{q\_m}^{2} \leq 10^{+110}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(2, p \cdot \frac{r}{t\_0}, \left|p\right|\right) + -4 \cdot \frac{q\_m \cdot q\_m}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ (fabs r) r)))
   (if (<= (pow q_m 2.0) 1e+110)
     (*
      (pow 2.0 -1.0)
      (+ (fma 2.0 (* p (/ r t_0)) (fabs p)) (* -4.0 (/ (* q_m q_m) t_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+110) {
		tmp = pow(2.0, -1.0) * (fma(2.0, (p * (r / t_0)), fabs(p)) + (-4.0 * ((q_m * q_m) / t_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)
	t_0 = Float64(abs(r) + r)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 1e+110)
		tmp = Float64((2.0 ^ -1.0) * Float64(fma(2.0, Float64(p * Float64(r / t_0)), abs(p)) + Float64(-4.0 * Float64(Float64(q_m * q_m) / t_0))));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]}, If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+110], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(2.0 * N[(p * N[(r / t$95$0), $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(q$95$m * q$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1e110

   1. Initial program 24.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-- N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in p around 0

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in q around 0

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

   10. Applied rewrites 34.8%

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

   11. Taylor expanded in q around 0

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

   13. Applied rewrites 34.7%

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

   if 1e110 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 23.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 34.8

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

   5. Applied rewrites 34.8%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{+110}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(2, p \cdot \frac{r}{\left|r\right| + r}, \left|p\right|\right) + -4 \cdot \frac{q \cdot q}{\left|r\right| + r}\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|r\right| - r\right) + \left|p\right|\right) + p\right)\\ \mathbf{elif}\;{q\_m}^{2} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 5e-164)
   (* 0.5 (+ (+ (- (fabs r) r) (fabs p)) p))
   (if (<= (pow q_m 2.0) 5e-18) (/ (* (- q_m) q_m) r) (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 5e-164) {
		tmp = 0.5 * (((fabs(r) - r) + fabs(p)) + p);
	} else if (pow(q_m, 2.0) <= 5e-18) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 5d-164) then
        tmp = 0.5d0 * (((abs(r) - r) + abs(p)) + p)
    else if ((q_m ** 2.0d0) <= 5d-18) then
        tmp = (-q_m * q_m) / r
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 5e-164) {
		tmp = 0.5 * (((Math.abs(r) - r) + Math.abs(p)) + p);
	} else if (Math.pow(q_m, 2.0) <= 5e-18) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 5e-164:
		tmp = 0.5 * (((math.fabs(r) - r) + math.fabs(p)) + p)
	elif math.pow(q_m, 2.0) <= 5e-18:
		tmp = (-q_m * q_m) / r
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 5e-164)
		tmp = Float64(0.5 * Float64(Float64(Float64(abs(r) - r) + abs(p)) + p));
	elseif ((q_m ^ 2.0) <= 5e-18)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 5e-164)
		tmp = 0.5 * (((abs(r) - r) + abs(p)) + p);
	elseif ((q_m ^ 2.0) <= 5e-18)
		tmp = (-q_m * q_m) / r;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-164], N[(0.5 * N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-18], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.99999999999999962e-164

   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 p around -inf

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

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 16.4

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

   5. Applied rewrites 16.4%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 39.8%

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

   if 4.99999999999999962e-164 < (pow.f64 q #s(literal 2 binary64)) < 5.00000000000000036e-18

   1. Initial program 7.2%

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

   3. Taylor expanded in r around inf

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 15.9%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 26.9%

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

   if 5.00000000000000036e-18 < (pow.f64 q #s(literal 2 binary64))

   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 29.7

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

   5. Applied rewrites 29.7%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|r\right| - r\right) + \left|p\right|\right) + p\right)\\ \mathbf{elif}\;{q}^{2} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 4 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left|r\right| - r\right) + \left|p\right|\right) + p\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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) 4e-46)
   (* 0.5 (+ (+ (- (fabs r) r) (fabs p)) p))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 4e-46) {
		tmp = 0.5 * (((fabs(r) - r) + fabs(p)) + p);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if ((q_m ** 2.0d0) <= 4d-46) then
        tmp = 0.5d0 * (((abs(r) - r) + abs(p)) + p)
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (Math.pow(q_m, 2.0) <= 4e-46) {
		tmp = 0.5 * (((Math.abs(r) - r) + Math.abs(p)) + p);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if math.pow(q_m, 2.0) <= 4e-46:
		tmp = 0.5 * (((math.fabs(r) - r) + math.fabs(p)) + 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) <= 4e-46)
		tmp = Float64(0.5 * Float64(Float64(Float64(abs(r) - r) + abs(p)) + p));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 4e-46)
		tmp = 0.5 * (((abs(r) - r) + abs(p)) + p);
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 4e-46], N[(0.5 * N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.00000000000000009e-46

   1. Initial program 23.0%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      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

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 14.2

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 34.2%

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

   if 4.00000000000000009e-46 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 24.6%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 28.9

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

   5. Applied rewrites 28.9%

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

Alternative 6: 36.1% 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 23.8%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 17.6

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

5. Applied rewrites 17.6%

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

Reproduce

herbie shell --seed 2024314 
(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)))))))