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

Percentage Accurate: 23.7% → 48.8%

Time: 9.6s

Alternatives: 9

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: 23.7% 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: 48.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left(\left|p\right| + p\right) + \left|r\right|\\ \mathbf{if}\;r \leq -2.15 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 4.25 \cdot 10^{+40}:\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{q\_m}{r} \cdot \frac{q\_m}{r}, \mathsf{fma}\left(\frac{t\_0}{r}, 0.5, -0.5\right)\right) \cdot r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (let* ((t_0 (+ (+ (fabs p) p) (fabs r))))
   (if (<= r -2.15e-175)
     (* (fma t_0 (/ 0.5 r) -0.5) r)
     (if (<= r 4.25e+40)
       (- q_m)
       (* (fma -1.0 (* (/ q_m r) (/ q_m r)) (fma (/ t_0 r) 0.5 -0.5)) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = (fabs(p) + p) + fabs(r);
	double tmp;
	if (r <= -2.15e-175) {
		tmp = fma(t_0, (0.5 / r), -0.5) * r;
	} else if (r <= 4.25e+40) {
		tmp = -q_m;
	} else {
		tmp = fma(-1.0, ((q_m / r) * (q_m / r)), fma((t_0 / r), 0.5, -0.5)) * r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(Float64(abs(p) + p) + abs(r))
	tmp = 0.0
	if (r <= -2.15e-175)
		tmp = Float64(fma(t_0, Float64(0.5 / r), -0.5) * r);
	elseif (r <= 4.25e+40)
		tmp = Float64(-q_m);
	else
		tmp = Float64(fma(-1.0, Float64(Float64(q_m / r) * Float64(q_m / r)), fma(Float64(t_0 / r), 0.5, -0.5)) * r);
	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[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -2.15e-175], N[(N[(t$95$0 * N[(0.5 / r), $MachinePrecision] + -0.5), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[r, 4.25e+40], (-q$95$m), N[(N[(-1.0 * N[(N[(q$95$m / r), $MachinePrecision] * N[(q$95$m / r), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / r), $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.14999999999999999e-175

   1. Initial program 21.3%

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

   3. Taylor expanded in 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 4.5%

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

   7. Applied rewrites 9.8%

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

   if -2.14999999999999999e-175 < r < 4.24999999999999998e40

   1. Initial program 31.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 24.5

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

   5. Applied rewrites 24.5%

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

   if 4.24999999999999998e40 < r

   1. Initial program 8.7%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{p \cdot {q}^{2}}{{r}^{3}} + \left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right)\right) - \frac{1}{2}\right)} \]

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 38.0%

      \[\leadsto \mathsf{fma}\left(-1, \frac{q}{r} \cdot \frac{q}{r}, \mathsf{fma}\left(\frac{\left(\left|r\right| + p\right) + \left|p\right|}{r}, 0.5, -0.5\right)\right) \cdot r \]
   9. Step-by-step derivation

   10. Applied rewrites 40.1%

       \[\leadsto \mathsf{fma}\left(-1, \frac{q}{r} \cdot \frac{q}{r}, \mathsf{fma}\left(\frac{\left(\left|p\right| + p\right) + \left|r\right|}{r}, 0.5, -0.5\right)\right) \cdot r \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 46.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{q\_m}\\ \mathbf{elif}\;{q\_m}^{2} \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + p\right) + \left|r\right|, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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) 0.0)
   (/ (* (- q_m) q_m) q_m)
   (if (<= (pow q_m 2.0) 5e-54)
     (* (fma (+ (+ (fabs p) p) (fabs r)) (/ 0.5 r) -0.5) 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) <= 0.0) {
		tmp = (-q_m * q_m) / q_m;
	} else if (pow(q_m, 2.0) <= 5e-54) {
		tmp = fma(((fabs(p) + p) + fabs(r)), (0.5 / r), -0.5) * 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) <= 0.0)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / q_m);
	elseif ((q_m ^ 2.0) <= 5e-54)
		tmp = Float64(fma(Float64(Float64(abs(p) + p) + abs(r)), Float64(0.5 / r), -0.5) * 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], 0.0], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / q$95$m), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-54], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(0.5 / r), $MachinePrecision] + -0.5), $MachinePrecision] * r), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 0.0

   1. Initial program 31.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 9.3

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

   5. Applied rewrites 9.3%

      \[\leadsto \color{blue}{-q} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0 - q \cdot q}{\color{blue}{0 + q}} \]

   if 0.0 < (pow.f64 q #s(literal 2 binary64)) < 5.00000000000000015e-54

   1. Initial program 15.9%

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

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 10.2%

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

   7. Applied rewrites 13.8%

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

   if 5.00000000000000015e-54 < (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 28.1

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

   5. Applied rewrites 28.1%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 0:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{q}\\ \mathbf{elif}\;{q}^{2} \leq 5 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + p\right) + \left|r\right|, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left(-q\_m\right) \cdot q\_m\\ \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-266}:\\ \;\;\;\;\frac{t\_0}{q\_m}\\ \mathbf{elif}\;{q\_m}^{2} \leq 10^{-36}:\\ \;\;\;\;\frac{t\_0 \cdot \left(r + p\right)}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (- q_m) q_m)))
   (if (<= (pow q_m 2.0) 5e-266)
     (/ t_0 q_m)
     (if (<= (pow q_m 2.0) 1e-36) (/ (* t_0 (+ r p)) (* 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 t_0 = -q_m * q_m;
	double tmp;
	if (pow(q_m, 2.0) <= 5e-266) {
		tmp = t_0 / q_m;
	} else if (pow(q_m, 2.0) <= 1e-36) {
		tmp = (t_0 * (r + p)) / (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) :: t_0
    real(8) :: tmp
    t_0 = -q_m * q_m
    if ((q_m ** 2.0d0) <= 5d-266) then
        tmp = t_0 / q_m
    else if ((q_m ** 2.0d0) <= 1d-36) then
        tmp = (t_0 * (r + p)) / (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 t_0 = -q_m * q_m;
	double tmp;
	if (Math.pow(q_m, 2.0) <= 5e-266) {
		tmp = t_0 / q_m;
	} else if (Math.pow(q_m, 2.0) <= 1e-36) {
		tmp = (t_0 * (r + p)) / (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):
	t_0 = -q_m * q_m
	tmp = 0
	if math.pow(q_m, 2.0) <= 5e-266:
		tmp = t_0 / q_m
	elif math.pow(q_m, 2.0) <= 1e-36:
		tmp = (t_0 * (r + p)) / (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)
	t_0 = Float64(Float64(-q_m) * q_m)
	tmp = 0.0
	if ((q_m ^ 2.0) <= 5e-266)
		tmp = Float64(t_0 / q_m);
	elseif ((q_m ^ 2.0) <= 1e-36)
		tmp = Float64(Float64(t_0 * Float64(r + p)) / Float64(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)
	t_0 = -q_m * q_m;
	tmp = 0.0;
	if ((q_m ^ 2.0) <= 5e-266)
		tmp = t_0 / q_m;
	elseif ((q_m ^ 2.0) <= 1e-36)
		tmp = (t_0 * (r + p)) / (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_] := Block[{t$95$0 = N[((-q$95$m) * q$95$m), $MachinePrecision]}, If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 5e-266], N[(t$95$0 / q$95$m), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-36], N[(N[(t$95$0 * N[(r + p), $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.99999999999999992e-266

   1. Initial program 29.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 11.1

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

   5. Applied rewrites 11.1%

      \[\leadsto \color{blue}{-q} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.8%

      \[\leadsto \frac{0 - q \cdot q}{\color{blue}{0 + q}} \]

   if 4.99999999999999992e-266 < (pow.f64 q #s(literal 2 binary64)) < 9.9999999999999994e-37

   1. Initial program 15.1%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{p \cdot {q}^{2}}{{r}^{3}} + \left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right)\right) - \frac{1}{2}\right)} \]

   5. Applied rewrites 15.0%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 19.3%

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

   if 9.9999999999999994e-37 < (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 29.8

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

   5. Applied rewrites 29.8%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{-266}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{q}\\ \mathbf{elif}\;{q}^{2} \leq 10^{-36}:\\ \;\;\;\;\frac{\left(\left(-q\right) \cdot q\right) \cdot \left(r + p\right)}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;{q\_m}^{2} \leq 5 \cdot 10^{-266}:\\ \;\;\;\;\frac{\left(-q\_m\right) \cdot q\_m}{q\_m}\\ \mathbf{elif}\;{q\_m}^{2} \leq 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\left(p + \left|r\right|\right) + \left|p\right|, 0.5, -0.5 \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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) 5e-266)
   (/ (* (- q_m) q_m) q_m)
   (if (<= (pow q_m 2.0) 1e-52)
     (fma (+ (+ p (fabs r)) (fabs p)) 0.5 (* -0.5 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) <= 5e-266) {
		tmp = (-q_m * q_m) / q_m;
	} else if (pow(q_m, 2.0) <= 1e-52) {
		tmp = fma(((p + fabs(r)) + fabs(p)), 0.5, (-0.5 * 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) <= 5e-266)
		tmp = Float64(Float64(Float64(-q_m) * q_m) / q_m);
	elseif ((q_m ^ 2.0) <= 1e-52)
		tmp = fma(Float64(Float64(p + abs(r)) + abs(p)), 0.5, Float64(-0.5 * 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], 5e-266], N[(N[((-q$95$m) * q$95$m), $MachinePrecision] / q$95$m), $MachinePrecision], If[LessEqual[N[Power[q$95$m, 2.0], $MachinePrecision], 1e-52], N[(N[(N[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-0.5 * r), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 q #s(literal 2 binary64)) < 4.99999999999999992e-266

   1. Initial program 29.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 11.1

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

   5. Applied rewrites 11.1%

      \[\leadsto \color{blue}{-q} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.8%

      \[\leadsto \frac{0 - q \cdot q}{\color{blue}{0 + q}} \]

   if 4.99999999999999992e-266 < (pow.f64 q #s(literal 2 binary64)) < 1e-52

   1. Initial program 13.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 11.8%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 11.9%

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

   if 1e-52 < (pow.f64 q #s(literal 2 binary64))

   1. Initial program 22.5%

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

   3. Taylor expanded in q around inf

      \[\leadsto \color{blue}{-1 \cdot q} \]
   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 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{q}^{2} \leq 5 \cdot 10^{-266}:\\ \;\;\;\;\frac{\left(-q\right) \cdot q}{q}\\ \mathbf{elif}\;{q}^{2} \leq 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(\left(p + \left|r\right|\right) + \left|p\right|, 0.5, -0.5 \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;-q\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -2.15 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + p\right) + \left|r\right|, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 2.05 \cdot 10^{+134}:\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{q\_m}{r} \cdot \frac{q\_m}{r}, \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{r}, 0.5, -0.5\right)\right) \cdot r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r -2.15e-175)
   (* (fma (+ (+ (fabs p) p) (fabs r)) (/ 0.5 r) -0.5) r)
   (if (<= r 2.05e+134)
     (- q_m)
     (*
      (fma
       -1.0
       (* (/ q_m r) (/ q_m r))
       (fma (/ (+ (fabs r) (fabs p)) r) 0.5 -0.5))
      r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= -2.15e-175) {
		tmp = fma(((fabs(p) + p) + fabs(r)), (0.5 / r), -0.5) * r;
	} else if (r <= 2.05e+134) {
		tmp = -q_m;
	} else {
		tmp = fma(-1.0, ((q_m / r) * (q_m / r)), fma(((fabs(r) + fabs(p)) / r), 0.5, -0.5)) * r;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= -2.15e-175)
		tmp = Float64(fma(Float64(Float64(abs(p) + p) + abs(r)), Float64(0.5 / r), -0.5) * r);
	elseif (r <= 2.05e+134)
		tmp = Float64(-q_m);
	else
		tmp = Float64(fma(-1.0, Float64(Float64(q_m / r) * Float64(q_m / r)), fma(Float64(Float64(abs(r) + abs(p)) / r), 0.5, -0.5)) * r);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, -2.15e-175], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(0.5 / r), $MachinePrecision] + -0.5), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[r, 2.05e+134], (-q$95$m), N[(N[(-1.0 * N[(N[(q$95$m / r), $MachinePrecision] * N[(q$95$m / r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.14999999999999999e-175

   1. Initial program 21.3%

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

   3. Taylor expanded in 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 4.5%

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

   7. Applied rewrites 9.8%

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

   if -2.14999999999999999e-175 < r < 2.0500000000000002e134

   1. Initial program 29.6%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 23.2

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

   5. Applied rewrites 23.2%

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

   if 2.0500000000000002e134 < r

   1. Initial program 5.2%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{p \cdot {q}^{2}}{{r}^{3}} + \left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right)\right) - \frac{1}{2}\right)} \]

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 40.1%

       \[\leadsto \mathsf{fma}\left(-1, \frac{q}{r} \cdot \frac{q}{r}, \mathsf{fma}\left(\frac{\left|r\right| + \left|p\right|}{r}, 0.5, -0.5\right)\right) \cdot r \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 48.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq -2.15 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left|p\right| + p\right) + \left|r\right|, \frac{0.5}{r}, -0.5\right) \cdot r\\ \mathbf{elif}\;r \leq 4.25 \cdot 10^{+40}:\\ \;\;\;\;-q\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-q\_m\right) \cdot q\_m\right) \cdot \frac{\frac{p}{r} + 1}{r}\\ \end{array} \end{array} \]

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 -2.15e-175)
   (* (fma (+ (+ (fabs p) p) (fabs r)) (/ 0.5 r) -0.5) r)
   (if (<= r 4.25e+40) (- q_m) (* (* (- q_m) q_m) (/ (+ (/ p r) 1.0) 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 <= -2.15e-175) {
		tmp = fma(((fabs(p) + p) + fabs(r)), (0.5 / r), -0.5) * r;
	} else if (r <= 4.25e+40) {
		tmp = -q_m;
	} else {
		tmp = (-q_m * q_m) * (((p / r) + 1.0) / 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 <= -2.15e-175)
		tmp = Float64(fma(Float64(Float64(abs(p) + p) + abs(r)), Float64(0.5 / r), -0.5) * r);
	elseif (r <= 4.25e+40)
		tmp = Float64(-q_m);
	else
		tmp = Float64(Float64(Float64(-q_m) * q_m) * Float64(Float64(Float64(p / r) + 1.0) / r));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, -2.15e-175], N[(N[(N[(N[(N[Abs[p], $MachinePrecision] + p), $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(0.5 / r), $MachinePrecision] + -0.5), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[r, 4.25e+40], (-q$95$m), N[(N[((-q$95$m) * q$95$m), $MachinePrecision] * N[(N[(N[(p / r), $MachinePrecision] + 1.0), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.14999999999999999e-175

   1. Initial program 21.3%

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

   3. Taylor expanded in 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 4.5%

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

   7. Applied rewrites 9.8%

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

   if -2.14999999999999999e-175 < r < 4.24999999999999998e40

   1. Initial program 31.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 24.5

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

   5. Applied rewrites 24.5%

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

   if 4.24999999999999998e40 < r

   1. Initial program 8.7%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\left(-1 \cdot \frac{p \cdot {q}^{2}}{{r}^{3}} + \left(-1 \cdot \frac{{q}^{2}}{{r}^{2}} + \frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r}\right)\right) - \frac{1}{2}\right)} \]

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in q around inf

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

   8. Applied rewrites 36.1%

      \[\leadsto \left(\left(-q\right) \cdot q\right) \cdot \color{blue}{\frac{\frac{p}{r} + 1}{r}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 41.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left(p + \left|r\right|\right) + \left|p\right|\\ \mathbf{if}\;q\_m \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;q\_m \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, -0.5 \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 (+ (+ p (fabs r)) (fabs p))))
   (if (<= q_m 2.2e-205)
     (* t_0 0.5)
     (if (<= q_m 7.2e-27) (fma t_0 0.5 (* -0.5 r)) (- 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 = (p + fabs(r)) + fabs(p);
	double tmp;
	if (q_m <= 2.2e-205) {
		tmp = t_0 * 0.5;
	} else if (q_m <= 7.2e-27) {
		tmp = fma(t_0, 0.5, (-0.5 * 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)
	t_0 = Float64(Float64(p + abs(r)) + abs(p))
	tmp = 0.0
	if (q_m <= 2.2e-205)
		tmp = Float64(t_0 * 0.5);
	elseif (q_m <= 7.2e-27)
		tmp = fma(t_0, 0.5, Float64(-0.5 * 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_] := Block[{t$95$0 = N[(N[(p + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 2.2e-205], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[q$95$m, 7.2e-27], N[(t$95$0 * 0.5 + N[(-0.5 * r), $MachinePrecision]), $MachinePrecision], (-q$95$m)]]]

Derivation

1. Split input into 3 regimes

2. if q < 2.20000000000000009e-205

   1. Initial program 24.6%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r} - \frac{1}{2}\right)} \]
   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 9.2%

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

   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 9.6%

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

   if 2.20000000000000009e-205 < q < 7.1999999999999997e-27

   1. Initial program 15.2%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 10.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 10.7%

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

   if 7.1999999999999997e-27 < q

   1. Initial program 22.6%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

Alternative 8: 40.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9 \cdot 10^{-67}:\\ \;\;\;\;\left(\left(p + \left|r\right|\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-q\_m\\ \end{array} \end{array} \]

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 9e-67) (* (+ (+ 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 <= 9e-67) {
		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 <= 9d-67) 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 <= 9e-67) {
		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 <= 9e-67:
		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 <= 9e-67)
		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 <= 9e-67)
		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, 9e-67], 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 < 9.00000000000000031e-67

   1. Initial program 23.7%

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

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{r \cdot \left(\frac{1}{2} \cdot \frac{\left(\left|p\right| + \left|r\right|\right) - -1 \cdot p}{r} - \frac{1}{2}\right)} \]
   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 9.1%

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

   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 9.3%

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

   if 9.00000000000000031e-67 < q

   1. Initial program 21.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   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 48.9

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

   5. Applied rewrites 48.9%

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

Alternative 9: 35.4% 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 22.8%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) - \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
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 20.1

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

5. Applied rewrites 20.1%

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

Reproduce

herbie shell --seed 2024302 
(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)))))))