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

Percentage Accurate: 24.1% → 57.8%

Time: 9.2s

Alternatives: 7

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: 24.1% 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: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{elif}\;{q\_m}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left|r\right| + p\right) + \left|p\right|}{r}, 0.5, -0.5\right), \frac{r}{q\_m \cdot q\_m}, \frac{-1}{r}\right) \cdot \left(q\_m \cdot q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (<= (pow q_m 2.0) 1e-108)
   (* (fma (/ (+ (fabs p) (- (fabs r) r)) p) -0.5 -0.5) (- p))
   (if (<= (pow q_m 2.0) 2e+31)
     (*
      (fma
       (fma (/ (+ (+ (fabs r) p) (fabs p)) r) 0.5 -0.5)
       (/ r (* q_m q_m))
       (/ -1.0 r))
      (* q_m q_m))
     (- 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-108) {
		tmp = fma(((fabs(p) + (fabs(r) - r)) / p), -0.5, -0.5) * -p;
	} else if (pow(q_m, 2.0) <= 2e+31) {
		tmp = fma(fma((((fabs(r) + p) + fabs(p)) / r), 0.5, -0.5), (r / (q_m * q_m)), (-1.0 / r)) * (q_m * q_m);
	} 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-108)
		tmp = Float64(fma(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p), -0.5, -0.5) * Float64(-p));
	elseif ((q_m ^ 2.0) <= 2e+31)
		tmp = Float64(fma(fma(Float64(Float64(Float64(abs(r) + p) + abs(p)) / r), 0.5, -0.5), Float64(r / Float64(q_m * q_m)), Float64(-1.0 / r)) * Float64(q_m * q_m));
	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-108], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5 + -0.5), $MachinePrecision] * (-p)), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+31], N[(N[(N[(N[(N[(N[(N[Abs[r], $MachinePrecision] + p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(r / N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / r), $MachinePrecision]), $MachinePrecision] * N[(q$95$m * q$95$m), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000004e-108

   1. Initial program 22.7%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 5.9

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

   5. Applied rewrites 5.9%

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

   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. mul-1-neg N/A

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

      3. lower-*.f64 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

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

   8. Applied rewrites 29.4%

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

   if 1.00000000000000004e-108 < (pow.f64 q #s(literal 2 binary64)) < 1.9999999999999999e31

   1. Initial program 18.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right) - \frac{1}{2}\right)} \]
   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 11.3%

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

   6. Taylor expanded in q around inf

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

   8. Applied rewrites 20.1%

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

   if 1.9999999999999999e31 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 25.8%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.2

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

   5. Applied rewrites 28.2%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{elif}\;{q}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left|r\right| + p\right) + \left|p\right|}{r}, 0.5, -0.5\right), \frac{r}{q \cdot q}, \frac{-1}{r}\right) \cdot \left(q \cdot q\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{elif}\;{q\_m}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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-108)
   (* (fma (/ (+ (fabs p) (- (fabs r) r)) p) -0.5 -0.5) (- p))
   (if (<= (pow q_m 2.0) 2e+31) (/ (* (- q_m) q_m) 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 (pow(q_m, 2.0) <= 1e-108) {
		tmp = fma(((fabs(p) + (fabs(r) - r)) / p), -0.5, -0.5) * -p;
	} else if (pow(q_m, 2.0) <= 2e+31) {
		tmp = (-q_m * q_m) / 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 ^ 2.0) <= 1e-108)
		tmp = Float64(fma(Float64(Float64(abs(p) + Float64(abs(r) - r)) / p), -0.5, -0.5) * Float64(-p));
	elseif ((q_m ^ 2.0) <= 2e+31)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / 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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-108], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - r), $MachinePrecision]), $MachinePrecision] / p), $MachinePrecision] * -0.5 + -0.5), $MachinePrecision] * (-p)), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+31], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000004e-108

   1. Initial program 22.7%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 5.9

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

   5. Applied rewrites 5.9%

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

   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. mul-1-neg N/A

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

      3. lower-*.f64 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

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

   8. Applied rewrites 29.4%

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

   if 1.00000000000000004e-108 < (pow.f64 q #s(literal 2 binary64)) < 1.9999999999999999e31

   1. Initial program 18.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right) - \frac{1}{2}\right)} \]
   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 11.3%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 19.2%

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

   if 1.9999999999999999e31 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 25.8%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.2

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

   5. Applied rewrites 28.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left|p\right| + \left(\left|r\right| - r\right)}{p}, -0.5, -0.5\right) \cdot \left(-p\right)\\ \mathbf{elif}\;{q}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 10^{-108}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;{q\_m}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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-108)
   (* (+ (fabs p) p) (/ 1.0 2.0))
   (if (<= (pow q_m 2.0) 2e+31) (/ (* (- q_m) q_m) 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 (pow(q_m, 2.0) <= 1e-108) {
		tmp = (fabs(p) + p) * (1.0 / 2.0);
	} else if (pow(q_m, 2.0) <= 2e+31) {
		tmp = (-q_m * q_m) / 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 ** 2.0d0) <= 1d-108) then
        tmp = (abs(p) + p) * (1.0d0 / 2.0d0)
    else if ((q_m ** 2.0d0) <= 2d+31) then
        tmp = (-q_m * q_m) / 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 (Math.pow(q_m, 2.0) <= 1e-108) {
		tmp = (Math.abs(p) + p) * (1.0 / 2.0);
	} else if (Math.pow(q_m, 2.0) <= 2e+31) {
		tmp = (-q_m * q_m) / 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 math.pow(q_m, 2.0) <= 1e-108:
		tmp = (math.fabs(p) + p) * (1.0 / 2.0)
	elif math.pow(q_m, 2.0) <= 2e+31:
		tmp = (-q_m * q_m) / 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 ^ 2.0) <= 1e-108)
		tmp = Float64(Float64(abs(p) + p) * Float64(1.0 / 2.0));
	elseif ((q_m ^ 2.0) <= 2e+31)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / 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 ^ 2.0) <= 1e-108)
		tmp = (abs(p) + p) * (1.0 / 2.0);
	elseif ((q_m ^ 2.0) <= 2e+31)
		tmp = (-q_m * q_m) / 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[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-108], N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 2e+31], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 1.00000000000000004e-108

   1. Initial program 22.7%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fabs.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 16.2

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

   5. Applied rewrites 16.2%

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

   7. Applied rewrites 11.7%

      \[\leadsto \frac{1}{2} \cdot \left(\left(\frac{\left|p\right|}{r} + \left({1}^{1.5} - \left(1 - \frac{p}{r}\right)\right)\right) \cdot r\right) \]

   8. Taylor expanded in p around 0

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

   10. Applied rewrites 29.7%

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

   if 1.00000000000000004e-108 < (pow.f64 q #s(literal 2 binary64)) < 1.9999999999999999e31

   1. Initial program 18.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right) - \frac{1}{2}\right)} \]
   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 11.3%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 19.2%

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

   if 1.9999999999999999e31 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 25.8%

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

   3. Taylor expanded in q around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.2

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

   5. Applied rewrites 28.2%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 10^{-108}:\\ \;\;\;\;\left(\left|p\right| + p\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;{q}^{2} \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(\left|p\right| + \left|r\right|\right) - 2 \cdot q\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 6.2e+31)
   (* (- (+ (fabs p) (fabs r)) (* 2.0 q_m)) 0.5)
   (/ (* (- q_m) q_m) r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 6.2e+31) {
		tmp = ((fabs(p) + fabs(r)) - (2.0 * q_m)) * 0.5;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 6.2d+31) then
        tmp = ((abs(p) + abs(r)) - (2.0d0 * q_m)) * 0.5d0
    else
        tmp = (-q_m * q_m) / r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 6.2e+31) {
		tmp = ((Math.abs(p) + Math.abs(r)) - (2.0 * q_m)) * 0.5;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 6.2e+31:
		tmp = ((math.fabs(p) + math.fabs(r)) - (2.0 * q_m)) * 0.5
	else:
		tmp = (-q_m * q_m) / r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 6.2e+31)
		tmp = Float64(Float64(Float64(abs(p) + abs(r)) - Float64(2.0 * q_m)) * 0.5);
	else
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 6.2e+31)
		tmp = ((abs(p) + abs(r)) - (2.0 * q_m)) * 0.5;
	else
		tmp = (-q_m * q_m) / r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 6.2e+31], N[(N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] - N[(2.0 * q$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.2000000000000004e31

   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

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

      1. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left|r\right| + \left|p\right|\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(q \cdot 2\right)\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   7. Applied rewrites 19.3%

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

   if 6.2000000000000004e31 < r

   1. Initial program 11.5%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right) - \frac{1}{2}\right)} \]
   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 23.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\left(-q\right) \cdot q}{\color{blue}{r}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(\left|p\right| + \left|r\right|\right) - 2 \cdot q\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 8.2 \cdot 10^{+31}:\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{r}\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 8.2e+31) (- q_m) (/ (* (- q_m) q_m) r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 8.2e+31) {
		tmp = -q_m;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Fortran

q_m = abs(q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
real(8) function code(p, r, q_m)
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 8.2d+31) then
        tmp = -q_m
    else
        tmp = (-q_m * q_m) / r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 8.2e+31) {
		tmp = -q_m;
	} else {
		tmp = (-q_m * q_m) / r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 8.2e+31:
		tmp = -q_m
	else:
		tmp = (-q_m * q_m) / r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 8.2e+31)
		tmp = Float64(-q_m);
	else
		tmp = Float64(Float64(Float64(-q_m) * q_m) / r);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 8.2e+31)
		tmp = -q_m;
	else
		tmp = (-q_m * q_m) / r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 8.2e+31], (-q$95$m), N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.2000000000000003e31

   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 19.7

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

   5. Applied rewrites 19.7%

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

   if 8.2000000000000003e31 < r

   1. Initial program 11.5%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right) - \frac{1}{2}\right)} \]
   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 23.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\left(-q\right) \cdot q}{\color{blue}{r}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 40.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;\left(\left(\left|r\right| + p\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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-105) (* (+ (+ (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-105) {
		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-105) 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-105) {
		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-105:
		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-105)
		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-105)
		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-105], 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 2 regimes

2. if q < 5.50000000000000029e-105

   1. Initial program 24.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r} - \frac{1}{2}\right)} \]
   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 8.4%

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

   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 5.7%

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

   if 5.50000000000000029e-105 < q

   1. Initial program 22.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 44.1

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

   5. Applied rewrites 44.1%

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

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;q \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;\left(\left(\left|r\right| + p\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% 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 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 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 18.5

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

5. Applied rewrites 18.5%

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

Reproduce

herbie shell --seed 2024295 
(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)))))))