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

Percentage Accurate: 44.9% → 82.5%

Time: 7.1s

Alternatives: 10

Speedup: 17.9×

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: 44.9% 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: 82.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5.8 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(\left|p\right| - \left(p - \left|r\right|\right), 0.5, r \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

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 5.8e+123)
   (fma (- (fabs p) (- p (fabs r))) 0.5 (* 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 <= 5.8e+123) {
		tmp = fma((fabs(p) - (p - fabs(r))), 0.5, (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 <= 5.8e+123)
		tmp = fma(Float64(abs(p) - Float64(p - abs(r))), 0.5, Float64(r * 0.5));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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, 5.8e+123], N[(N[(N[Abs[p], $MachinePrecision] - N[(p - N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(r * 0.5), $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 < 5.80000000000000019e123

   1. Initial program 50.0%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 43.7%

      \[\leadsto \left(\left(\left(\left|r\right| + r\right) + \left|p\right|\right) - p\right) \cdot \color{blue}{0.5} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \left(\left(\left|p\right| - p\right) + \left(\left|r\right| + r\right)\right) \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 44.5%

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

   if 5.80000000000000019e123 < q

   1. Initial program 15.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 91.0%

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

Alternative 2: 59.8% accurate, 2.0× speedup

Math

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

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 (<= (pow q_m 2.0) 1e+94)
   (* (+ (+ (fabs r) r) (fabs p)) 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 (pow(q_m, 2.0) <= 1e+94) {
		tmp = ((fabs(r) + r) + fabs(p)) * 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 ^ 2.0) <= 1e+94)
		tmp = Float64(Float64(Float64(abs(r) + r) + abs(p)) * 0.5);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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+94], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $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 (pow.f64 q #s(literal 2 binary64)) < 1e94

   1. Initial program 59.4%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 45.5

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 36.6%

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

   if 1e94 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 25.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 43.2%

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

Alternative 3: 65.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;p \leq -2.65 \cdot 10^{+51}:\\ \;\;\;\;\left(t\_0 - p\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -3.15 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (+ (fabs r) (fabs p))))
   (if (<= p -2.65e+51)
     (* (- t_0 p) 0.5)
     (if (<= p -3.15e-260)
       (fma 0.5 t_0 q_m)
       (* (+ (+ (fabs r) 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 = fabs(r) + fabs(p);
	double tmp;
	if (p <= -2.65e+51) {
		tmp = (t_0 - p) * 0.5;
	} else if (p <= -3.15e-260) {
		tmp = fma(0.5, t_0, q_m);
	} else {
		tmp = ((fabs(r) + 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(abs(r) + abs(p))
	tmp = 0.0
	if (p <= -2.65e+51)
		tmp = Float64(Float64(t_0 - p) * 0.5);
	elseif (p <= -3.15e-260)
		tmp = fma(0.5, t_0, q_m);
	else
		tmp = Float64(Float64(Float64(abs(r) + 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[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, -2.65e+51], N[(N[(t$95$0 - p), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, -3.15e-260], N[(0.5 * t$95$0 + q$95$m), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -2.6499999999999998e51

   1. Initial program 25.5%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 74.3%

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

   if -2.6499999999999998e51 < p < -3.14999999999999989e-260

   1. Initial program 48.8%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 36.5%

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

   if -3.14999999999999989e-260 < p

   1. Initial program 50.1%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 15.9

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

   5. Applied rewrites 15.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 26.5%

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

Alternative 4: 82.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5.8 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(\left|p\right| - p\right) + \left(\left|r\right| + r\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

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 5.8e+123)
   (* (+ (- (fabs p) 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 <= 5.8e+123) {
		tmp = ((fabs(p) - 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 <= 5.8e+123)
		tmp = Float64(Float64(Float64(abs(p) - p) + Float64(abs(r) + r)) * 0.5);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), 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, 5.8e+123], N[(N[(N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision]), $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 < 5.80000000000000019e123

   1. Initial program 50.0%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-fabs.f64 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 43.7%

      \[\leadsto \left(\left(\left(\left|r\right| + r\right) + \left|p\right|\right) - p\right) \cdot \color{blue}{0.5} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

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

   if 5.80000000000000019e123 < q

   1. Initial program 15.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 91.0%

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

Alternative 5: 38.8% accurate, 13.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.95 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \left(\left|r\right| + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 1.95e-94) (* 0.5 (+ (fabs r) (fabs p))) (* 1.0 q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.95e-94) {
		tmp = 0.5 * (fabs(r) + fabs(p));
	} else {
		tmp = 1.0 * q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 1.95d-94) then
        tmp = 0.5d0 * (abs(r) + abs(p))
    else
        tmp = 1.0d0 * q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.95e-94) {
		tmp = 0.5 * (Math.abs(r) + Math.abs(p));
	} else {
		tmp = 1.0 * q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 1.95e-94:
		tmp = 0.5 * (math.fabs(r) + math.fabs(p))
	else:
		tmp = 1.0 * q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 1.95e-94)
		tmp = Float64(0.5 * Float64(abs(r) + abs(p)));
	else
		tmp = Float64(1.0 * q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 1.95e-94)
		tmp = 0.5 * (abs(r) + abs(p));
	else
		tmp = 1.0 * q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 1.95e-94], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 1.9500000000000001e-94

   1. Initial program 48.4%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 11.4

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 15.4%

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

   if 1.9500000000000001e-94 < q

   1. Initial program 36.2%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in q around inf

      \[\leadsto 1 \cdot q \]
   7. Step-by-step derivation

   8. Applied rewrites 56.7%

      \[\leadsto 1 \cdot q \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 45.0% accurate, 17.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right) \end{array} \]

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 (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) {
	return fma(0.5, (fabs(r) + fabs(p)), q_m);
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return fma(0.5, Float64(abs(r) + abs(p)), 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_] := N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

3. Taylor expanded in q around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-fabs.f64 N/A

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

   10. lower-fabs.f64 29.1

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

5. Applied rewrites 29.1%

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

6. Taylor expanded in q around 0

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

8. Applied rewrites 31.8%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]
9. Add Preprocessing

Alternative 7: 36.4% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.8 \cdot 10^{-123}:\\ \;\;\;\;0.5 \cdot r\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 3.8e-123) (* 0.5 r) (* 1.0 q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.8e-123) {
		tmp = 0.5 * r;
	} else {
		tmp = 1.0 * 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 <= 3.8d-123) then
        tmp = 0.5d0 * r
    else
        tmp = 1.0d0 * q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.8e-123) {
		tmp = 0.5 * r;
	} else {
		tmp = 1.0 * q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 3.8e-123:
		tmp = 0.5 * r
	else:
		tmp = 1.0 * q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 3.8e-123)
		tmp = Float64(0.5 * r);
	else
		tmp = Float64(1.0 * 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 <= 3.8e-123)
		tmp = 0.5 * r;
	else
		tmp = 1.0 * q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 3.8e-123], N[(0.5 * r), $MachinePrecision], N[(1.0 * q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.79999999999999996e-123

   1. Initial program 47.9%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 6.4

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 6.4%

      \[\leadsto \color{blue}{0.5 \cdot r} \]

   if 3.79999999999999996e-123 < q

   1. Initial program 38.0%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. lower-fabs.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in q around inf

      \[\leadsto 1 \cdot q \]
   7. Step-by-step derivation

   8. Applied rewrites 53.8%

      \[\leadsto 1 \cdot q \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 12.9% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

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 (<= p -1.2e+39) (* -0.5 p) (* 0.5 r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -1.2e+39) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	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 (p <= (-1.2d+39)) then
        tmp = (-0.5d0) * p
    else
        tmp = 0.5d0 * r
    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 (p <= -1.2e+39) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	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 p <= -1.2e+39:
		tmp = -0.5 * p
	else:
		tmp = 0.5 * r
	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 (p <= -1.2e+39)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(0.5 * r);
	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 (p <= -1.2e+39)
		tmp = -0.5 * p;
	else
		tmp = 0.5 * r;
	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[p, -1.2e+39], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -1.2e39

   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 p around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
   4. Step-by-step derivation

      1. lower-*.f64 15.1

         \[\leadsto \color{blue}{-0.5 \cdot p} \]

   5. Applied rewrites 15.1%

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

   if -1.2e39 < p

   1. Initial program 49.3%

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

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 6.5

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 6.5%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 8.6% accurate, 41.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ -0.5 \cdot p \end{array} \]

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 (* -0.5 p))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return -0.5 * p;
}

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 = (-0.5d0) * p
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 -0.5 * p;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return -0.5 * p

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-0.5 * p)
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = -0.5 * p;
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_] := N[(-0.5 * p), $MachinePrecision]

Derivation

1. Initial program 44.3%

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

3. Taylor expanded in p around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot p} \]
4. Step-by-step derivation

   1. lower-*.f64 5.5

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

5. Applied rewrites 5.5%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
6. Add Preprocessing

Alternative 10: 1.2% accurate, 83.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ -q\_m \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (- q_m))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return -q_m;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = -q_m
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return -q_m;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return -q_m

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-q_m)
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = -q_m;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := (-q$95$m)

Derivation

1. Initial program 44.3%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 17.5

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

5. Applied rewrites 17.5%

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

Reproduce

herbie shell --seed 2024306 
(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)))))))