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

Percentage Accurate: 24.5% → 58.0%

Time: 9.2s

Alternatives: 4

Speedup: 83.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% 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: 58.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 20000000000000:\\ \;\;\;\;\left(-p\right) \cdot \mathsf{fma}\left(\frac{\left(\left|r\right| - r\right) + \left|p\right|}{p}, -0.5, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, -q\_m\right)\\ \end{array} \end{array} \]

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) 20000000000000.0)
   (* (- p) (fma (/ (+ (- (fabs r) r) (fabs p)) p) -0.5 -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) <= 20000000000000.0) {
		tmp = -p * fma((((fabs(r) - r) + fabs(p)) / p), -0.5, -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) <= 20000000000000.0)
		tmp = Float64(Float64(-p) * fma(Float64(Float64(Float64(abs(r) - r) + abs(p)) / p), -0.5, -0.5));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), Float64(-q_m));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 20000000000000.0], N[((-p) * N[(N[(N[(N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5 + -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)) < 2e13

   1. Initial program 23.2%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 6.9

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

   5. Applied rewrites 6.9%

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

   6. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-fabs.f64 28.8

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

   8. Applied rewrites 28.8%

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

   if 2e13 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.7%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 41.7%

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

Alternative 2: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 20000000000000:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, -q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 20000000000000.0)
   (/ (* (- q_m) q_m) r)
   (fma 0.5 (+ (fabs r) (fabs p)) (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 20000000000000.0) {
		tmp = (-q_m * q_m) / r;
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), -q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 20000000000000.0)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), Float64(-q_m));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 20000000000000.0], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $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)) < 2e13

   1. Initial program 23.2%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 37.1%

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

   if 2e13 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 27.7%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 41.7%

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

Alternative 3: 40.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 2 \cdot 10^{-232}:\\ \;\;\;\;\left(\left(p + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, -q\_m\right)\\ \end{array} \end{array} \]

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) 2e-232)
   (* (+ (+ p (fabs 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) <= 2e-232) {
		tmp = ((p + fabs(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) <= 2e-232)
		tmp = Float64(Float64(Float64(p + abs(r)) + abs(p)) * 0.5);
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), Float64(-q_m));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e-232], N[(N[(N[(p + N[Abs[r], $MachinePrecision]), $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)) < 2.00000000000000005e-232

   1. Initial program 28.9%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 11.7%

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

   if 2.00000000000000005e-232 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 23.9%

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

   3. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 33.2%

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

Alternative 4: 36.2% 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 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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 24.5

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

5. Applied rewrites 24.5%

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

Reproduce

herbie shell --seed 2024319 
(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)))))))