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

Percentage Accurate: 46.4% → 81.0%

Time: 6.9s

Alternatives: 10

Speedup: 11.4×

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: 46.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| - p\right) + r, 0.5, \left|r\right| \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 4e+139)
   (fma (+ (- (fabs p) p) r) 0.5 (* (fabs 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 (pow(q_m, 2.0) <= 4e+139) {
		tmp = fma(((fabs(p) - p) + r), 0.5, (fabs(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 ^ 2.0) <= 4e+139)
		tmp = fma(Float64(Float64(abs(p) - p) + r), 0.5, Float64(abs(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[N[Power[q$95$m, 2.0], $MachinePrecision], 4e+139], N[(N[(N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision] + r), $MachinePrecision] * 0.5 + N[(N[Abs[r], $MachinePrecision] * 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 (pow.f64 q #s(literal 2 binary64)) < 4.00000000000000013e139

   1. Initial program 51.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 44.5

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

   5. Applied rewrites 44.5%

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

   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 49.6%

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

   10. Applied rewrites 49.5%

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

   12. Applied rewrites 50.0%

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

   if 4.00000000000000013e139 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 29.5%

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

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   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 38.2

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

   5. Applied rewrites 38.2%

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

   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 38.2%

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

Alternative 2: 81.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 4 \cdot 10^{+139}:\\ \;\;\;\;\left(\left|r\right| + \left(r + \left(\left|p\right| - p\right)\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

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) 4e+139)
   (* (+ (fabs r) (+ r (- (fabs p) 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) <= 4e+139) {
		tmp = (fabs(r) + (r + (fabs(p) - 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) <= 4e+139)
		tmp = Float64(Float64(abs(r) + Float64(r + Float64(abs(p) - 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], 4e+139], N[(N[(N[Abs[r], $MachinePrecision] + N[(r + N[(N[Abs[p], $MachinePrecision] - p), $MachinePrecision]), $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)) < 4.00000000000000013e139

   1. Initial program 51.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 44.5

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

   5. Applied rewrites 44.5%

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

   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 49.6%

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

   10. Applied rewrites 49.5%

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

   if 4.00000000000000013e139 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 29.5%

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

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   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 38.2

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

   5. Applied rewrites 38.2%

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

   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 38.2%

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

Alternative 3: 38.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\left|r\right| + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 5e+39) (* 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 (pow(q_m, 2.0) <= 5e+39) {
		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 ** 2.0d0) <= 5d+39) 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 (Math.pow(q_m, 2.0) <= 5e+39) {
		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 math.pow(q_m, 2.0) <= 5e+39:
		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 ^ 2.0) <= 5e+39)
		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 ^ 2.0) <= 5e+39)
		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[N[Power[q$95$m, 2.0], $MachinePrecision], 5e+39], 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 (pow.f64 q #s(literal 2 binary64)) < 5.00000000000000015e39

   1. Initial program 51.5%

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

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   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 14.4

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

   5. Applied rewrites 14.4%

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

   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 18.4%

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

   if 5.00000000000000015e39 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 33.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in q around inf

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

   8. Applied rewrites 31.5%

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

Alternative 4: 65.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;r \leq -1.52 \cdot 10^{-202}:\\ \;\;\;\;\left(t\_0 - p\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(r + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= r -1.52e-202)
     (* (- t_0 p) 0.5)
     (if (<= r 1.7e+99)
       (fma 0.5 t_0 q_m)
       (* (+ (+ 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 = fabs(r) + fabs(p);
	double tmp;
	if (r <= -1.52e-202) {
		tmp = (t_0 - p) * 0.5;
	} else if (r <= 1.7e+99) {
		tmp = fma(0.5, t_0, q_m);
	} else {
		tmp = ((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(abs(r) + abs(p))
	tmp = 0.0
	if (r <= -1.52e-202)
		tmp = Float64(Float64(t_0 - p) * 0.5);
	elseif (r <= 1.7e+99)
		tmp = fma(0.5, t_0, q_m);
	else
		tmp = Float64(Float64(Float64(r + 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[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -1.52e-202], N[(N[(t$95$0 - p), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 1.7e+99], N[(0.5 * t$95$0 + q$95$m), $MachinePrecision], N[(N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < -1.5200000000000001e-202

   1. Initial program 38.2%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{\left|p\right| + \left(\left|r\right| + -1 \cdot p\right)}{r}\right)} \]
   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 8.9

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

   5. Applied rewrites 8.9%

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

   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 17.3%

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

   if -1.5200000000000001e-202 < r < 1.69999999999999992e99

   1. Initial program 56.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 30.9

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

   5. Applied rewrites 30.9%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 32.5%

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

   if 1.69999999999999992e99 < r

   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 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 87.3

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

   5. Applied rewrites 87.3%

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

   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 85.6%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 75.4%

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

Alternative 5: 60.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(r + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \end{array} \end{array} \]

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
 (if (<= r 1.7e+99)
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)
   (* (+ (+ 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 (r <= 1.7e+99) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = ((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 (r <= 1.7e+99)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(Float64(Float64(r + 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[r, 1.7e+99], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.69999999999999992e99

   1. Initial program 47.8%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 28.1

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

   5. Applied rewrites 28.1%

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

   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 30.2%

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

   if 1.69999999999999992e99 < r

   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 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 87.3

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

   5. Applied rewrites 87.3%

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

   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 85.6%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 75.4%

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

Alternative 6: 45.7% 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 42.8%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 25.6

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

5. Applied rewrites 25.6%

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

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 28.3%

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

Alternative 7: 36.9% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.2 \cdot 10^{+183}:\\ \;\;\;\;1 \cdot q\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

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 (<= r 1.2e+183) (* 1.0 q_m) (* 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 (r <= 1.2e+183) {
		tmp = 1.0 * q_m;
	} 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 (r <= 1.2d+183) then
        tmp = 1.0d0 * q_m
    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 (r <= 1.2e+183) {
		tmp = 1.0 * q_m;
	} 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 r <= 1.2e+183:
		tmp = 1.0 * q_m
	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 (r <= 1.2e+183)
		tmp = Float64(1.0 * q_m);
	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 (r <= 1.2e+183)
		tmp = 1.0 * q_m;
	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[r, 1.2e+183], N[(1.0 * q$95$m), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.2000000000000001e183

   1. Initial program 47.1%

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

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q \cdot \left(1 + \frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right)} \]
   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 27.1

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

   5. Applied rewrites 27.1%

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

   6. Taylor expanded in q around inf

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

   8. Applied rewrites 19.5%

      \[\leadsto 1 \cdot q \]

   if 1.2000000000000001e183 < r

   1. Initial program 7.9%

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

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

   5. Applied rewrites 17.6%

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

Alternative 8: 12.9% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.82 \cdot 10^{-34}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

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 (<= r 1.82e-34) (* -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 (r <= 1.82e-34) {
		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 (r <= 1.82d-34) 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 (r <= 1.82e-34) {
		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 r <= 1.82e-34:
		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 (r <= 1.82e-34)
		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 (r <= 1.82e-34)
		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[r, 1.82e-34], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.82000000000000009e-34

   1. Initial program 46.1%

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

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

   if 1.82000000000000009e-34 < r

   1. Initial program 35.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 13.0

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

   5. Applied rewrites 13.0%

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

Alternative 9: 8.5% 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 42.8%

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

3. Taylor expanded in p around -inf

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

   1. lower-*.f64 4.9

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

5. Applied rewrites 4.9%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
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 42.8%

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

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 19.2

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

5. Applied rewrites 19.2%

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

Reproduce

herbie shell --seed 2024322 
(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)))))))