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

Percentage Accurate: 24.0% → 69.5%

Time: 11.8s

Alternatives: 6

Speedup: 83.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.0% 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: 69.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| - r\\ \mathbf{if}\;q\_m \leq 3 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \mathsf{fma}\left(\frac{q\_m \cdot q\_m}{r}, -2, \left|p\right|\right)\right) + t\_0\right)\\ \mathbf{elif}\;q\_m \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(-2 \cdot q\_m, \frac{q\_m}{r}, \left|p\right| + p\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (- (fabs r) r)))
   (if (<= q_m 3e-29)
     (* 0.5 (+ (+ p (fma (/ (* q_m q_m) r) -2.0 (fabs p))) t_0))
     (if (<= q_m 2.2e+34)
       (* (pow 2.0 -1.0) (+ (fma (* -2.0 q_m) (/ q_m r) (+ (fabs p) p)) t_0))
       (- q_m)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) - r;
	double tmp;
	if (q_m <= 3e-29) {
		tmp = 0.5 * ((p + fma(((q_m * q_m) / r), -2.0, fabs(p))) + t_0);
	} else if (q_m <= 2.2e+34) {
		tmp = pow(2.0, -1.0) * (fma((-2.0 * q_m), (q_m / r), (fabs(p) + p)) + t_0);
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(r) - r)
	tmp = 0.0
	if (q_m <= 3e-29)
		tmp = Float64(0.5 * Float64(Float64(p + fma(Float64(Float64(q_m * q_m) / r), -2.0, abs(p))) + t_0));
	elseif (q_m <= 2.2e+34)
		tmp = Float64((2.0 ^ -1.0) * Float64(fma(Float64(-2.0 * q_m), Float64(q_m / r), Float64(abs(p) + p)) + t_0));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]}, If[LessEqual[q$95$m, 3e-29], N[(0.5 * N[(N[(p + N[(N[(N[(q$95$m * q$95$m), $MachinePrecision] / r), $MachinePrecision] * -2.0 + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[q$95$m, 2.2e+34], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(N[(-2.0 * q$95$m), $MachinePrecision] * N[(q$95$m / r), $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if q < 3.0000000000000003e-29

   1. Initial program 24.1%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 14.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 25.0%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 28.8%

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

       1. lift-/.f64 N/A

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

       2. metadata-eval 28.8

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

   13. Applied rewrites 28.8%

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

   if 3.0000000000000003e-29 < q < 2.2000000000000002e34

   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 r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 14.0%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 14.1%

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

   13. Applied rewrites 14.1%

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

   if 2.2000000000000002e34 < q

   1. Initial program 25.3%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.5

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

   5. Applied rewrites 64.5%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 3 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \mathsf{fma}\left(\frac{q \cdot q}{r}, -2, \left|p\right|\right)\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{elif}\;q \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;{2}^{-1} \cdot \left(\mathsf{fma}\left(-2 \cdot q, \frac{q}{r}, \left|p\right| + p\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;{2}^{-1} \cdot \left(\left(\left|p\right| + p\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 1.12e+18)
   (* (pow 2.0 -1.0) (+ (+ (fabs p) p) (- (fabs r) r)))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.12e+18) {
		tmp = pow(2.0, -1.0) * ((fabs(p) + p) + (fabs(r) - r));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 1.12d+18) then
        tmp = (2.0d0 ** (-1.0d0)) * ((abs(p) + p) + (abs(r) - r))
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.12e+18) {
		tmp = Math.pow(2.0, -1.0) * ((Math.abs(p) + p) + (Math.abs(r) - r));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 1.12e+18:
		tmp = math.pow(2.0, -1.0) * ((math.fabs(p) + p) + (math.fabs(r) - r))
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 1.12e+18)
		tmp = Float64((2.0 ^ -1.0) * Float64(Float64(abs(p) + p) + Float64(abs(r) - r)));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 1.12e+18)
		tmp = (2.0 ^ -1.0) * ((abs(p) + p) + (abs(r) - r));
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 1.12e+18], N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 1.12e18

   1. Initial program 23.6%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 13.5%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 24.6%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 28.1%

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

   12. Taylor expanded in r around inf

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

   14. Applied rewrites 25.9%

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

   if 1.12e18 < q

   1. Initial program 26.7%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.8

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

   5. Applied rewrites 62.8%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;{2}^{-1} \cdot \left(\left(\left|p\right| + p\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.75 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \left(\left(p + \mathsf{fma}\left(\frac{q\_m \cdot q\_m}{r}, -2, \left|p\right|\right)\right) + \left(\left|r\right| - r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 1.75e+29)
   (* 0.5 (+ (+ p (fma (/ (* q_m q_m) r) -2.0 (fabs p))) (- (fabs r) r)))
   (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.75e+29) {
		tmp = 0.5 * ((p + fma(((q_m * q_m) / r), -2.0, fabs(p))) + (fabs(r) - r));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 1.75e+29)
		tmp = Float64(0.5 * Float64(Float64(p + fma(Float64(Float64(q_m * q_m) / r), -2.0, abs(p))) + Float64(abs(r) - r)));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 1.75e+29], N[(0.5 * N[(N[(p + N[(N[(N[(q$95$m * q$95$m), $MachinePrecision] / r), $MachinePrecision] * -2.0 + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 1.74999999999999989e29

   1. Initial program 23.8%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 27.7%

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

       1. lift-/.f64 N/A

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

       2. metadata-eval 27.7

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

   13. Applied rewrites 27.7%

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

   if 1.74999999999999989e29 < q

   1. Initial program 26.2%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.0

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

   5. Applied rewrites 64.0%

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

Alternative 4: 43.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 1.75 \cdot 10^{-93}:\\ \;\;\;\;-0.5 \cdot \left(r - \left(\left(p + \left|r\right|\right) + \left|p\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 1.75e-93) (* -0.5 (- r (+ (+ p (fabs r)) (fabs p)))) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.75e-93) {
		tmp = -0.5 * (r - ((p + fabs(r)) + fabs(p)));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 1.75d-93) then
        tmp = (-0.5d0) * (r - ((p + abs(r)) + abs(p)))
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 1.75e-93) {
		tmp = -0.5 * (r - ((p + Math.abs(r)) + Math.abs(p)));
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 1.75e-93:
		tmp = -0.5 * (r - ((p + math.fabs(r)) + math.fabs(p)))
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 1.75e-93)
		tmp = Float64(-0.5 * Float64(r - Float64(Float64(p + abs(r)) + abs(p))));
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 1.75e-93)
		tmp = -0.5 * (r - ((p + abs(r)) + abs(p)));
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 1.75e-93], N[(-0.5 * N[(r - N[(N[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 1.75e-93

   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

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r} - \frac{1}{2}\right)} \]
   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 11.9%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 11.9%

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

   if 1.75e-93 < q

   1. Initial program 26.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 52.6

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

   5. Applied rewrites 52.6%

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

Alternative 5: 40.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.5 \cdot 10^{-166}:\\ \;\;\;\;\left(\left(p + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 3.5e-166) (* (+ (+ p (fabs r)) (fabs p)) 0.5) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.5e-166) {
		tmp = ((p + fabs(r)) + fabs(p)) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 3.5d-166) then
        tmp = ((p + abs(r)) + abs(p)) * 0.5d0
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.5e-166) {
		tmp = ((p + Math.abs(r)) + Math.abs(p)) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 3.5e-166:
		tmp = ((p + math.fabs(r)) + math.fabs(p)) * 0.5
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 3.5e-166)
		tmp = Float64(Float64(Float64(p + abs(r)) + abs(p)) * 0.5);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 3.5e-166)
		tmp = ((p + abs(r)) + abs(p)) * 0.5;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 3.5e-166], N[(N[(N[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 3.4999999999999999e-166

   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

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r} - \frac{1}{2}\right)} \]
   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 11.4%

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

   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 8.2%

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

   if 3.4999999999999999e-166 < q

   1. Initial program 26.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 48.0

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

   5. Applied rewrites 48.0%

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

Alternative 6: 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 24.3%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 19.5

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

5. Applied rewrites 19.5%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))