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

Percentage Accurate: 23.8% → 57.1%

Time: 8.3s

Alternatives: 5

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: 23.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left(\left|p\right| + p\right), 0.5, -0.5 \cdot r\right)\\ \mathbf{elif}\;{q\_m}^{2} \leq 10^{+141}:\\ \;\;\;\;\frac{q\_m \cdot q\_m}{p}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e-73)
   (fma (+ (fabs r) (+ (fabs p) p)) 0.5 (* -0.5 r))
   (if (<= (pow q_m 2.0) 1e+141) (/ (* q_m q_m) p) (- q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e-73) {
		tmp = fma((fabs(r) + (fabs(p) + p)), 0.5, (-0.5 * r));
	} else if (pow(q_m, 2.0) <= 1e+141) {
		tmp = (q_m * q_m) / 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 ^ 2.0) <= 1e-73)
		tmp = fma(Float64(abs(r) + Float64(abs(p) + p)), 0.5, Float64(-0.5 * r));
	elseif ((q_m ^ 2.0) <= 1e+141)
		tmp = Float64(Float64(q_m * q_m) / p);
	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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-73], N[(N[(N[Abs[r], $MachinePrecision] + N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-0.5 * r), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+141], N[(N[(q$95$m * q$95$m), $MachinePrecision] / p), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 9.99999999999999997e-74

   1. Initial program 28.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 8.6

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

   5. Applied rewrites 8.6%

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

   6. Taylor expanded in r around inf

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

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 16.5%

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

   13. Applied rewrites 36.9%

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

   if 9.99999999999999997e-74 < (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000002e141

   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 p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in p around inf

      \[\leadsto \frac{{q}^{2}}{p} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.1%

       \[\leadsto \frac{q \cdot q}{p} \]

   if 1.00000000000000002e141 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 22.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 32.3

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

   5. Applied rewrites 32.3%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(\left|r\right| + \left(\left|p\right| + p\right), 0.5, -0.5 \cdot r\right)\\ \mathbf{elif}\;{q}^{2} \leq 10^{+141}:\\ \;\;\;\;\frac{q \cdot q}{p}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{+141}:\\ \;\;\;\;\frac{q\_m \cdot q\_m}{p}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= (pow q_m 2.0) 1e+141) (/ (* q_m q_m) p) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (pow(q_m, 2.0) <= 1e+141) {
		tmp = (q_m * q_m) / 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 ** 2.0d0) <= 1d+141) then
        tmp = (q_m * q_m) / 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 (Math.pow(q_m, 2.0) <= 1e+141) {
		tmp = (q_m * q_m) / 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 math.pow(q_m, 2.0) <= 1e+141:
		tmp = (q_m * q_m) / 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 ^ 2.0) <= 1e+141)
		tmp = Float64(Float64(q_m * q_m) / 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 ^ 2.0) <= 1e+141)
		tmp = (q_m * q_m) / 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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e+141], N[(N[(q$95$m * q$95$m), $MachinePrecision] / p), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000002e141

   1. Initial program 27.2%

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

   3. Taylor expanded in p around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 14.6%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in p around inf

      \[\leadsto \frac{{q}^{2}}{p} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.5%

       \[\leadsto \frac{q \cdot q}{p} \]

   if 1.00000000000000002e141 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 22.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 32.3

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

   5. Applied rewrites 32.3%

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

Alternative 3: 40.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5.5 \cdot 10^{-173}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - r\right) \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 7 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\left|r\right| + p\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 5.5e-173)
   (* (- (+ (fabs r) (fabs p)) r) 0.5)
   (if (<= q_m 7e-60) (* (+ (+ (fabs r) p) (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 <= 5.5e-173) {
		tmp = ((fabs(r) + fabs(p)) - r) * 0.5;
	} else if (q_m <= 7e-60) {
		tmp = ((fabs(r) + p) + 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 <= 5.5d-173) then
        tmp = ((abs(r) + abs(p)) - r) * 0.5d0
    else if (q_m <= 7d-60) then
        tmp = ((abs(r) + p) + 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 <= 5.5e-173) {
		tmp = ((Math.abs(r) + Math.abs(p)) - r) * 0.5;
	} else if (q_m <= 7e-60) {
		tmp = ((Math.abs(r) + p) + 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 <= 5.5e-173:
		tmp = ((math.fabs(r) + math.fabs(p)) - r) * 0.5
	elif q_m <= 7e-60:
		tmp = ((math.fabs(r) + p) + 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 <= 5.5e-173)
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - r) * 0.5);
	elseif (q_m <= 7e-60)
		tmp = Float64(Float64(Float64(abs(r) + p) + 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 <= 5.5e-173)
		tmp = ((abs(r) + abs(p)) - r) * 0.5;
	elseif (q_m <= 7e-60)
		tmp = ((abs(r) + p) + 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, 5.5e-173], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 7e-60], N[(N[(N[(N[Abs[r], $MachinePrecision] + p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if q < 5.50000000000000022e-173

   1. Initial program 27.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in r around inf

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

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

   8. Applied rewrites 25.0%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 11.3%

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

   12. Taylor expanded in p around 0

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

   14. Applied rewrites 9.1%

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

   if 5.50000000000000022e-173 < q < 6.99999999999999952e-60

   1. Initial program 28.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 15.7

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in r around inf

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

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 10.2%

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

   if 6.99999999999999952e-60 < q

   1. Initial program 20.1%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.6

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

   5. Applied rewrites 49.6%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 5.5 \cdot 10^{-173}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - r\right) \cdot 0.5\\ \mathbf{elif}\;q \leq 7 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\left|r\right| + p\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 6.8 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(\left|r\right| + \left|p\right|\right) - r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 6.8e-69) (* (- (+ (fabs r) (fabs p)) r) 0.5) (- q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 6.8e-69) {
		tmp = ((fabs(r) + fabs(p)) - r) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (q_m <= 6.8d-69) then
        tmp = ((abs(r) + abs(p)) - r) * 0.5d0
    else
        tmp = -q_m
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 6.8e-69) {
		tmp = ((Math.abs(r) + Math.abs(p)) - r) * 0.5;
	} else {
		tmp = -q_m;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if q_m <= 6.8e-69:
		tmp = ((math.fabs(r) + math.fabs(p)) - r) * 0.5
	else:
		tmp = -q_m
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 6.8e-69)
		tmp = Float64(Float64(Float64(abs(r) + abs(p)) - r) * 0.5);
	else
		tmp = Float64(-q_m);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (q_m <= 6.8e-69)
		tmp = ((abs(r) + abs(p)) - r) * 0.5;
	else
		tmp = -q_m;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 6.8e-69], N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] - r), $MachinePrecision] * 0.5), $MachinePrecision], (-q$95$m)]

Derivation

1. Split input into 2 regimes

2. if q < 6.80000000000000016e-69

   1. Initial program 27.6%

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

   3. Taylor expanded in 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 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in r around inf

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

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in r around 0

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

   11. Applied rewrites 11.9%

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

   12. Taylor expanded in p around 0

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

   14. Applied rewrites 9.8%

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

   if 6.80000000000000016e-69 < q

   1. Initial program 19.9%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

Alternative 5: 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 18.2

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

5. Applied rewrites 18.2%

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

Reproduce

herbie shell --seed 2024284 
(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)))))))