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

Percentage Accurate: 24.6% → 57.7%

Time: 9.3s

Alternatives: 8

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: 24.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left(\left(p + \left|p\right|\right) + \left|r\right|\right) - r\\ \mathbf{if}\;q\_m \leq 1.45 \cdot 10^{-206}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, 0.5, 0.5\right) \cdot p\\ \mathbf{elif}\;q\_m \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 33000000000:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\frac{p}{r \cdot r} \cdot -2 - \frac{2}{r}, q\_m \cdot q\_m, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, q\_m, \mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q\_m}, 0.5, \frac{r}{q\_m} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- (+ (+ p (fabs p)) (fabs r)) r)))
   (if (<= q_m 1.45e-206)
     (* (fma (/ (+ (- (fabs r) r) (fabs p)) p) 0.5 0.5) p)
     (if (<= q_m 7.5e-67)
       (* t_0 0.5)
       (if (<= q_m 33000000000.0)
         (* 0.5 (fma (- (* (/ p (* r r)) -2.0) (/ 2.0 r)) (* q_m q_m) t_0))
         (*
          (fma
           -2.0
           q_m
           (fma
            (fma (/ p q_m) 0.5 (* (/ r q_m) -0.25))
            r
            (+ (fabs r) (fabs p))))
          0.5))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = ((p + fabs(p)) + fabs(r)) - r;
	double tmp;
	if (q_m <= 1.45e-206) {
		tmp = fma((((fabs(r) - r) + fabs(p)) / p), 0.5, 0.5) * p;
	} else if (q_m <= 7.5e-67) {
		tmp = t_0 * 0.5;
	} else if (q_m <= 33000000000.0) {
		tmp = 0.5 * fma((((p / (r * r)) * -2.0) - (2.0 / r)), (q_m * q_m), t_0);
	} else {
		tmp = fma(-2.0, q_m, fma(fma((p / q_m), 0.5, ((r / q_m) * -0.25)), r, (fabs(r) + fabs(p)))) * 0.5;
	}
	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(Float64(Float64(p + abs(p)) + abs(r)) - r)
	tmp = 0.0
	if (q_m <= 1.45e-206)
		tmp = Float64(fma(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p), 0.5, 0.5) * p);
	elseif (q_m <= 7.5e-67)
		tmp = Float64(t_0 * 0.5);
	elseif (q_m <= 33000000000.0)
		tmp = Float64(0.5 * fma(Float64(Float64(Float64(p / Float64(r * r)) * -2.0) - Float64(2.0 / r)), Float64(q_m * q_m), t_0));
	else
		tmp = Float64(fma(-2.0, q_m, fma(fma(Float64(p / q_m), 0.5, Float64(Float64(r / q_m) * -0.25)), r, Float64(abs(r) + abs(p)))) * 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_] := Block[{t$95$0 = N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision]}, If[LessEqual[q$95$m, 1.45e-206], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * p), $MachinePrecision], If[LessEqual[q$95$m, 7.5e-67], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 33000000000.0], N[(0.5 * N[(N[(N[(N[(p / N[(r * r), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - N[(2.0 / r), $MachinePrecision]), $MachinePrecision] * N[(q$95$m * q$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * q$95$m + N[(N[(N[(p / q$95$m), $MachinePrecision] * 0.5 + N[(N[(r / q$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * r + N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if q < 1.4500000000000001e-206

   1. Initial program 31.7%

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

   3. Taylor expanded in 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 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in p around inf

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

   8. Applied rewrites 20.4%

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

   if 1.4500000000000001e-206 < q < 7.5000000000000005e-67

   1. Initial program 22.9%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 37.1%

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

   if 7.5000000000000005e-67 < q < 3.3e10

   1. Initial program 25.7%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 14.3%

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

   if 3.3e10 < q

   1. Initial program 17.6%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 14.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 58.7%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 59.3%

       \[\leadsto \mathsf{fma}\left(-2, q, \mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q}, 0.5, \frac{r}{q} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right)\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 40.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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) 4e-193) (* (- (+ (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) <= 4e-193) {
		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) <= 4d-193) 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) <= 4e-193) {
		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) <= 4e-193:
		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) <= 4e-193)
		tmp = Float64(Float64(Float64(abs(r) + 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) <= 4e-193)
		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], 4e-193], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.0000000000000002e-193

   1. Initial program 27.0%

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

   3. Taylor expanded in 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 17.3

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

   5. Applied rewrites 17.3%

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

   6. Taylor expanded in r around -inf

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

   8. Applied rewrites 13.2%

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

   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 14.2%

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

   if 4.0000000000000002e-193 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 24.7

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

   5. Applied rewrites 24.7%

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

Alternative 3: 56.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.4 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, 0.5, 0.5\right) \cdot p\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, q\_m, \mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q\_m}, 0.5, \frac{r}{q\_m} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= q_m 3.4e-49)
   (* (fma (/ (+ (- (fabs r) r) (fabs p)) p) 0.5 0.5) p)
   (*
    (fma
     -2.0
     q_m
     (fma (fma (/ p q_m) 0.5 (* (/ r q_m) -0.25)) r (+ (fabs r) (fabs p))))
    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 (q_m <= 3.4e-49) {
		tmp = fma((((fabs(r) - r) + fabs(p)) / p), 0.5, 0.5) * p;
	} else {
		tmp = fma(-2.0, q_m, fma(fma((p / q_m), 0.5, ((r / q_m) * -0.25)), r, (fabs(r) + fabs(p)))) * 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 <= 3.4e-49)
		tmp = Float64(fma(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p), 0.5, 0.5) * p);
	else
		tmp = Float64(fma(-2.0, q_m, fma(fma(Float64(p / q_m), 0.5, Float64(Float64(r / q_m) * -0.25)), r, Float64(abs(r) + abs(p)))) * 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[q$95$m, 3.4e-49], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * p), $MachinePrecision], N[(N[(-2.0 * q$95$m + N[(N[(N[(p / q$95$m), $MachinePrecision] * 0.5 + N[(N[(r / q$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * r + N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.40000000000000005e-49

   1. Initial program 30.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 10.9

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

   5. Applied rewrites 10.9%

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

   6. Taylor expanded in p around inf

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

   8. Applied rewrites 23.2%

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

   if 3.40000000000000005e-49 < q

   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 p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - q \cdot 2\right) \cdot 0.5 \]

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 53.3%

       \[\leadsto \mathsf{fma}\left(-2, q, \mathsf{fma}\left(\mathsf{fma}\left(\frac{p}{q}, 0.5, \frac{r}{q} \cdot -0.25\right), r, \left|r\right| + \left|p\right|\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 54.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.45 \cdot 10^{-206}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, 0.5, 0.5\right) \cdot p\\ \mathbf{elif}\;q\_m \leq 6 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\left(p + \left|p\right|\right) + \left|r\right|\right) - r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, -q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 1.45e-206)
   (* (fma (/ (+ (- (fabs r) r) (fabs p)) p) 0.5 0.5) p)
   (if (<= q_m 6e-94)
     (* (- (+ (+ p (fabs p)) (fabs r)) r) 0.5)
     (fma 0.5 (+ (fabs r) (fabs p)) (- q_m)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.45e-206) {
		tmp = fma((((fabs(r) - r) + fabs(p)) / p), 0.5, 0.5) * p;
	} else if (q_m <= 6e-94) {
		tmp = (((p + fabs(p)) + fabs(r)) - r) * 0.5;
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), -q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 1.45e-206)
		tmp = Float64(fma(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p), 0.5, 0.5) * p);
	elseif (q_m <= 6e-94)
		tmp = Float64(Float64(Float64(Float64(p + abs(p)) + abs(r)) - r) * 0.5);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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[q$95$m, 1.45e-206], N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * p), $MachinePrecision], If[LessEqual[q$95$m, 6e-94], N[(N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + (-q$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if q < 1.4500000000000001e-206

   1. Initial program 31.7%

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

   3. Taylor expanded in 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 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in p around inf

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

   8. Applied rewrites 20.4%

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

   if 1.4500000000000001e-206 < q < 6.0000000000000003e-94

   1. Initial program 11.2%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 4.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 37.6%

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

   if 6.0000000000000003e-94 < q

   1. Initial program 23.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 50.9%

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

Alternative 5: 55.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 6 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\left(p + \left|p\right|\right) + \left|r\right|\right) - r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, -q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 6e-94)
   (* (- (+ (+ p (fabs p)) (fabs r)) r) 0.5)
   (fma 0.5 (+ (fabs r) (fabs p)) (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 6e-94) {
		tmp = (((p + fabs(p)) + fabs(r)) - r) * 0.5;
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), -q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 6e-94)
		tmp = Float64(Float64(Float64(Float64(p + abs(p)) + abs(r)) - r) * 0.5);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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[q$95$m, 6e-94], N[(N[(N[(N[(p + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + (-q$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 6.0000000000000003e-94

   1. Initial program 29.1%

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

   3. Taylor expanded in p around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 18.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}}, r \cdot p, \left(\left|r\right| + \left|p\right|\right) - \sqrt{\mathsf{fma}\left(q \cdot q, 4, r \cdot r\right)}\right) \cdot 0.5} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 22.4%

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

   if 6.0000000000000003e-94 < q

   1. Initial program 23.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 50.9%

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

Alternative 6: 46.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 960000000000:\\ \;\;\;\;\left(-q\_m\right) \cdot \frac{q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, -q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 960000000000.0)
   (* (- q_m) (/ q_m r))
   (fma 0.5 (+ (fabs r) (fabs p)) (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 960000000000.0) {
		tmp = -q_m * (q_m / r);
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), -q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 960000000000.0)
		tmp = Float64(Float64(-q_m) * Float64(q_m / r));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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[q$95$m, 960000000000.0], N[((-q$95$m) * N[(q$95$m / r), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + (-q$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 9.6e11

   1. Initial program 29.7%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 12.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 26.0%

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

   if 9.6e11 < q

   1. Initial program 18.2%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 61.0%

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

Alternative 7: 40.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\left(t\_0 - r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, -q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= q_m 7.6e-96) (* (- t_0 r) 0.5) (fma 0.5 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) + fabs(p);
	double tmp;
	if (q_m <= 7.6e-96) {
		tmp = (t_0 - r) * 0.5;
	} else {
		tmp = fma(0.5, t_0, -q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(r) + abs(p))
	tmp = 0.0
	if (q_m <= 7.6e-96)
		tmp = Float64(Float64(t_0 - r) * 0.5);
	else
		tmp = fma(0.5, t_0, 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] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 7.6e-96], N[(N[(t$95$0 - r), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * t$95$0 + (-q$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 7.6000000000000001e-96

   1. Initial program 29.3%

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

   3. Taylor expanded in p around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(p \cdot \left(\frac{-1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - r}{p} - \frac{1}{2}\right)\right)} \]
   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 10.7

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

   5. Applied rewrites 10.7%

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

   6. Taylor expanded in r around -inf

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

   8. Applied rewrites 8.3%

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

   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 8.6%

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

   if 7.6000000000000001e-96 < q

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 50.3%

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

Alternative 8: 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 27.4%

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

3. Taylor expanded in q around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 17.5

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

5. Applied rewrites 17.5%

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

Reproduce

herbie shell --seed 2024323 
(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)))))))